High-order arbitrary Lagrangian–Eulerian discontinuous Galerkinmethods for the incompressible Navier–Stokes equations

Niklas Fehn∗, Johannes Heinz, Wolfgang A. Wall, Martin Kronbichler

Institute for Computational Mechanics, Technical University of Munich,Boltzmannstr. 15, 85748 Garching, Germany

Abstract

This paper presents robust discontinuous Galerkin methods for the incompressible Navier–Stokes equationson moving meshes. High-order accurate arbitrary Lagrangian–Eulerian formulations are proposed in a unifiedframework for both monolithic as well as projection or splitting-type Navier–Stokes solvers. The frameworkis flexible, allows implicit and explicit formulations of the convective term, and adaptive time-stepping. TheNavier–Stokes equations with ALE transport term are solved on the deformed geometry storing one instanceof the mesh that is updated from one time step to the next. Discretization in space is applied to the timediscrete equations so that all weak forms and mass matrices are evaluated at the end of the current time step.This design ensures that the proposed formulations fulfill the geometric conservation law automatically, as isshown theoretically and demonstrated numerically by the example of the free-stream preservation test. Wediscuss the peculiarities related to the imposition of boundary conditions in intermediate steps of projection-type methods and the ingredients needed to preserve high-order accuracy. We show numerically that theformulations proposed in this work maintain the formal order of accuracy of the Navier–Stokes solvers.Moreover, we demonstrate robustness and accuracy for under-resolved turbulent flows.

Keywords: arbitrary Lagrangian–Eulerian (ALE), incompressible Navier–Stokes, discontinuous Galerkin,matrix-free methods, projection methods

1. Introduction

The arbitrary Lagrangian–Eulerian (ALE) continuum mechanics description is the basis of many methodsto capture flow problems on deforming domains. A very prominent class of applications are fluid–structureinteraction problems with moderate deformations of the structure, where moderate means that a mesh mov-ing or mesh smoothing algorithm is able to handle the mesh deformation of the fluid mesh following thedeformations imposed at the fluid–structure interface, as opposed to very large deformations and topologicalchanges that require other, geometrically more flexible techniques. ALE methods have a long tradition andhave first been developed for finite difference methods [1], see also [2] for a review on the early developmentof this methodology, as well as [3] for a survey of ALE methods. It has later been developed for finite elementdiscretizations of the compressible Navier–Stokes equations in [4, 2], of the incompressible Navier–Stokesequations using linear elements in [5] and spectral element discretizations in [6], and also for finite volumediscretizations, see for example [7]. In the context of discontinuous Galerkin (DG) discretizations, thistechnique has first been applied to the compressible Navier–Stokes equations being solved on the deformingdomain [8, 9, 10], or solving transformed equations on a reference domain [11, 12]. For the incompressible

∗Corresponding author at: Institute for Computational Mechanics, Technical University of Munich, Boltzmannstr. 15,85748 Garching, Germany. Tel.: +49 89 28915300; fax: +49 89 28915301

Email addresses: fehn@lnm.mw.tum.de (Niklas Fehn), johannes.heinz@tum.de (Johannes Heinz), wall@lnm.mw.tum.de(Wolfgang A. Wall), kronbichler@lnm.mw.tum.de (Martin Kronbichler)

Preprint submitted to Journal March 17, 2020

arX

iv:2

003.

0716

6v1

[ph

ysic

s.co

mp-

ph]

16

Mar

202

0

Navier–Stokes equations, the development of ALE-DG methods lagged somewhat behind as detailed below.An ALE method satisfying the geometric conservation law (GCL) [13] is able to preserve a constant flowstate on moving meshes. The GCL has extensively been discussed in the context of finite volume discretiza-tions of the compressible Navier–Stokes equations, see for example the review article [14] and referencestherein. Here, we especially refer to two works [15, 16] addressing the solution of incompressible flow prob-lems and being particularly relevant for the present work. Following the design described in [15] one caneasily construct incompressible flow solvers that automatically fulfill the geometric conservation law, andwe therefore do not discuss this topic at length in the present work.

Before discussing ALE-DG methods for incompressible flows, let us first summarize the key issue ofincompressible Navier–Stokes DG solvers in the Eulerian case with static meshes. The main problem orig-inates from the nonlinearity of the convective term in case the numerical velocity field is not pointwisedivergence-free like with standard L2-conforming spaces. In this circumstance, the energy stability derivede.g. for an upwind flux with linear transport terms of constant speed is lost. Special DG discretizationsthat are exactly mass conserving and energy stable have been discussed in early mathematical literatureon DG methods for the incompressible Navier–Stokes equations, see [17, 18, 19]. However, the importanceof this aspect and its relevance for practical problems have long not been realized in application-orientedDG literature, where no attempts have been made to fulfill these properties [20, 21, 22, 23, 24, 25]. It waslater found in [26, 27, 28, 29] that compliance with (i) the divergence-free constraint and (ii) inter-elementcontinuity of the normal velocity is crucial in obtaining robust solvers for engineering applications wherethe solution is under-resolved, such as turbulent flows. Methods explicitly addressing these two require-ments can be categorized into two groups, those fulfilling them exactly for example by choosing appropriatefunction spaces such as H(div)-conforming spaces where the divergence of the velocity space lies in thepressure space [17, 18, 19, 30, 31, 32, 33], and those fulfilling them weakly by appropriate stabilizationterms [27, 28, 29, 34, 35]. These constraints on the velocity solution can be imposed in form of a postpro-cessing step or in an inbuilt/monolithic way, independently of the two categories a method belongs to. Acomparative study of the two categories has been shown recently in [36] in the context of under-resolvedturbulent flows.

ALE formulations for the incompressible Navier–Stokes equations using DG discretizations have beenpresented in [37, 38], with the mesh motion being restricted to rigid body rotations without mesh deformationin [37]. Both methods are based on the dual splitting projection scheme proposed by [39, 40] and use adiscontinuous Galerkin discretization of the velocity–pressure coupling terms without integration by parts,originating from the method proposed in [22], that has been shown to be unstable for small time stepsizes by independent studies, see [41, 42]. Both works [37, 38] use equal-order poylnomials for velocity andpressure, but it was shown in [41] that the dual splitting scheme per se is not inf–sup stable as is sometimesbelieved. Morever, the works [37, 38] enforce the important aspect of H(div)-conformity discussed aboveneither exactly nor weakly, so that the robustness of these methods remains questionable. These worksalso do not comment on the fulfillment of the geometric conservation law theoretically or by numericalexperiments. Second-order convergence in time is shown in [37] for a rotating, non-deforming mesh withfixed boundaries, hence, not being representative of a deforming fluid domain or a fluid–structure interactionproblem. A temporal convergence test is shown in [38] for a full FSI problem by comparing the error againstthe solution for the smallest time step size and second-order accuracy is shown. It remains unclear fromthese works whether third order accuracy can be achieved for the dual splitting scheme in the ALE case,which has for example been shown in [28] for a DG method solving the Eulerian form of the equations.Open questions remain therefore from previous works [37, 38] on ALE-DG methods for the incompressibleNavier–Stokes equations regarding how these methods are implemented exactly and questions about theirnumerical properties in terms of stability (small time steps, inf–sup problem, and under-resolved turbulenceor energy stability), fulfillment of the geometric conservation law, and temporal convergence rates.

More sophisticated convergence tests are presented in [43] for a space–time HDG method on deformingdomains, where it is argued that the reasons for choosing the space–time approach are automatically satifyingthe geometric conservation law and achieving arbitrarily high order in space and time. An energy-stableversion of such a space–time HDG approach has been proposed recently in [44] achieved through a velocityfield that is H(div)-conforming and exactly divergence-free. As shown in the present work, fulfillment

2

of the GCL and discretizations that formally exhibit arbitrarily high order of accuracy in space can alsobe achieved with a classical method-of-lines approach. Achieving arbitrarily high order of accuracy intime with projection-type Navier–Stokes solvers is non-trivial, irrespective of the grid motion. However,our experience is that second or third-order time integration schemes are sufficient in terms of accuracyfor practical problems, especially if the time step size is restricted according to the CFL condition whentreating the convective term explicitly in time, which is the state-of-the-art solution technique used by someof the most sophisticated and computationally efficient high-order CFD solvers, such as Nektar++[45] andNek5000 [46]. We emphasize that formal orders of accuracy describe the optimal behavior observable onlyin the asymptotic regime for sufficiently smooth solution. In this context, it should be mentioned that highorder of convergence in space is rarely observed for practical problems. The dissipation/dispersion propertiesand the associated improved resolution capabilities of high-order methods rather than theoretical rates ofconvergence motivate the use of high-order methods for application-relevant, turbulent flows [47, 48, 49].When it comes to the aspect of computational costs, it has not yet been demonstrated that space–timeapproaches can keep up with the fast solution techniques for incompressible flows mentioned above. Asimilar argument holds for the computational efficiency of matrix-based HDG solvers, which significantlytrail behind fast matrix-free DG implementations on modern CPU hardware, as shown by a recent study [50].

For these reasons, our goal is to develop ALE-DG methods based on the method-of-lines approach thatcombine computationally efficient matrix-free DG implementations [51] and fast Navier–Stokes solution al-gorithms [41, 29] with desirable discretization properties in terms of optimal convergence rates in time andspace, the geometric conservation law, and robustness for turbulent flows. To develop algorithms as simpleas possible, we make the following design choices: We solve the ALE equations on the deformed geometry,storing one instance of the mesh that is updated from one time step to the next by updating the coordinatesof all nodal points. We introduce the ALE equations on the level of differential equations, subsequentlydiscretized in time and space. This way, it is straight-forward to satisfy the geometric conservation lawautomatically [15], i.e., independently of the mesh motion and how the mesh velocity is computed numeri-cally. We present a unified framework for both monolithic solvers and widely used projection-type solvers.The formulation is flexible regarding implicit versus explicit formulations of the convective term, and theframework naturally includes the option for adaptive time-stepping. Although we consider analytical meshmotions in the present work, the methods are formulated with fluid–structure interaction problems in mind,i.e., the ALE formulations are implemented in a way that they only require knowledge about the coordinatesof all grid nodes at discrete instances of time, and the grid velocity is computed from these grid coordinatesin a way that the formal order of accuracy of the time integration schemes is maintained on moving meshes.

The outline of this article is as follows. We derive the ALE form of the incompressible Navier–Stokesequations in Section 2. Aspects related to the temporal discretization are discussed in Section 3, and thespatial discretization is subject of Section 4. Here, our focus lies on the aspects relevant to ALE, with thegoal to provide a comprehensive and simple formulation that can easily be included in existing flow solvers.Numerical results are presented in Section 5, and we summarize our results in Section 6.

2. Incompressible Navier–Stokes equations in arbitrary Lagrangian–Eulerian formulation

We consider the incompressible Navier–Stokes equations in a domain Ω ⊂ Rd

∂u

∂t

∣∣∣∣X

−∇ · Fv(u) +∇p = f , (1)

∇ · u = 0 , (2)

where u = (u1, ..., ud)T is the velocity vector and p the kinematic pressure. The body force vector is denoted

by f = (f1, ..., fd)T. Spatial derivatives are defined w.r.t. the Eulerian coordinates x = (x1, ..., xd)

T, ∇φ =∂φ∂x , and |X denotes the material time derivative, i.e., the total time derivative along the trajectory of amaterial point X of the fluid. The viscous term is written in Laplace formulation Fv(u) = ν∇u with theconstant kinematic viscosity ν.To obtain the ALE form of the above equations, the material time derivative (or the Eulerian time derivative

3

when considering the incompressible Navier–Stokes equations written in Eulerian coordinates) is replacedby a time derivative with respect to a fixed point of the mesh (denoted as ALE time derivative in thefollowing), which gives rise to an additional transport term with transport by the grid velocity uG. However,this transport term has the same structure as the convective term in Eulerian description, so that the sameimplementation with transport velocity w = u − uG instead of the fluid velocity u can be used. Themotivation behind is to apply the time integration scheme with an update of the solution vectors just asin the Eulerian case, thereby automatically obtaining the solution coefficients on the new mesh. By thistechnique, expensive transformations of the solution vector (containing the degrees of freedom of the finiteelement expansion) from one mesh at a previous time instant onto another one at the current time instantis avoided. Apart from the Eulerian coordinates x describing an (arbitrary) point in Euclidean space andthe material coordinates X describing a fixed fluid element, we introduce the mesh coordinates χ describinga fixed point of the mesh. For transient problems, x(X, t) and x(χ, t) describe the trajectories of a fixedfluid element or a fixed point of the mesh in the Eulerian coordinates x as a function of time. The fluid andmesh velocity are therefore given as

u =∂x

∂t

∣∣∣∣X

, uG =∂x

∂t

∣∣∣∣χ

.

The material time derivative of an arbitrary quantity φ is given as

∂φ (x(X, t), t)

∂t

∣∣∣∣X

=∂φ

∂t

∣∣∣∣x

+∂φ

∂x· ∂x∂t

∣∣∣∣X︸ ︷︷ ︸

=u

=∂φ

∂t

∣∣∣∣x

+ (u · ∇)φ . (3)

The same relation can be stated for the mesh reference frame χ to obtain the desired relation between theEulerian and ALE time derivatives

∂φ (x(χ, t), t)

∂t

∣∣∣∣χ

=∂φ

∂t

∣∣∣∣x

+∂φ

∂x· ∂x∂t

∣∣∣∣χ︸ ︷︷ ︸

=uG

=∂φ

∂t

∣∣∣∣x

+ (uG · ∇)φ . (4)

Inserting equations (3) and (4) into equation (1), we arrive at the incompressible Navier–Stokes equationsin ALE formulation

∂u

∂t

∣∣∣∣χ

+ ((u− uG) · ∇)u−∇ · Fv(u) +∇p = f in Ω(t) , (5)

∇ · u = 0 in Ω(t) . (6)

Remark Note that the formulation chosen as a starting point for discretization in time and space hasimportant implications regarding compliance with the geometric conservation law. Using the above differ-ential formulation with the convective term written in non-conservative form to derive temporal and spatialdiscretization allows to satisfy the GCL automatically [15]. Alternative conservative formulations with timederivative in front of the integral over a temporally changing domain contain an additional term in whichthe divergence of the mesh velocity occurs, and fulfilling the GCL is more complicated in this case [15, 16].We mention that the study [16] is inconclusive in the sense that the non-conservative formulation is notrestricted to first-order accuracy in time as implied in that work. As demonstrated in [15] and in the presentwork, high-order accuracy in time can be achieved with the non-conservative formulation.

The motion of the domain Ω(t) is described by a function fG = fG(χ, t)

fG :

Ω0 × [0, T ]→ Ω(t), Ω0,Ω(t) ⊂ Rd ,(χ, t) 7→ x (χ, t) .

4

ξ

χ

x

fG(χ, t)

fm(ξ, t)

fm(ξ, t = 0)

Ωe

Ωe(t = 0)

Ωe(t)

Ω0

Ω(t)

Figure 1: Illustration of coordinate systems χ and x with mesh deformation fG, and reference coordinates ξ with finite elementmapping fm of polynomial degree km = 2.

With respect to the argument χ, the map fG is a homeomorphism for all times, and the argument tdescribes a continuous deformation over time. In the context of this work and for the numerical resultsshown below, fG will be an analytically defined, smooth function in space and time. An illustration isshown in Figure 1. Without loss of generality, we assume x (χ, t = 0) = χ, and therefore Ω(t = 0) = Ω0.To demonstrate high order of accuracy of the multistep BDF time integration schemes that are used inthis work and that require a starting procedure to demonstrate the formal order of accuracy, it is essentialthat the mesh motion is continuously differentiable in time. In the context of fluid–structure interaction,the mesh motion is defined by the deformation of the fluid–structure interface according to the structuraldisplacements and a mesh smoothing algorithm calculating the mesh deformation in the interior of the fluiddomain.

The incompressible Navier–Stokes equations (5) and (6) are subject to the initial condition

u(x, t = 0) = u0(x) in Ω0 ,

where u0(x) is divergence-free and fulfills the velocity Dirichlet boundary condition shown below. On theboundary Γ = ∂Ω = ΓD∪ΓN with ΓD∩ΓN = ∅, Dirichlet and Neumann boundary conditions are prescribed

u = gu on ΓD(t) , (7)

(Fv(u)− pI) · n = h on ΓN(t) , (8)

where n is the outward pointing unit normal vector and I the identity matrix. As explained in [41], theNeumann boundary condition is split into a viscous part hu and a pressure part gp in case of projection-typeNavier–Stokes solvers, i.e., Fv(u) · n = hu and p = gp on ΓN(t) with h = hu − gpn. Another characteristicof the incompressible Navier–Stokes equations is that the pressure is only defined up to an additive constantin case of pure Dirichlet boundary conditions, Γ = ΓD. A unique pressure solution is obtained from theconstraint

∫Ωp dΩ = 0. In that special case the velocity Dirichlet boundary condition has to fulfill the

constraint∫

ΓD gu · n dΓ = 0, which is the integral version of the continuity equation (6), transformed intoa surface integral via Gauss’ divergence theorem.

3. Temporal discretization

A multitude of solution strategies have been proposed over the last decades to solve the incompressibleNavier–Stokes equations. While monolithic approaches are straight-forward in terms of time integration and

5

in achieving high-order accuracy, projection methods with a splitting of velocity and pressure unknowns areparticularly interesting from the point of view of computational costs, as these techniques decompose theproblem into easier-to-solve equations such as simple Poisson or Helmholtz-like problems. The literature onprojection methods is vast, see [52, 53] for an overview. Here, we focus on those methods that we believeare most widely used and that also cover different aspects related to implicit versus mixed explicit–implicitformulations and the availability of high-order formulations, e.g., through rotational formulations. As rep-resentatives of projection methods, we investigate (incremental) pressure-correction schemes in rotationalform, see [54, 55, 56, 57, 58, 59] for the development of this approach, as well as velocity-correction schemes,using the high-order formulation proposed in [39, 40]. For these methods, the splitting of the incompressibleNavier–Stokes equations is performed on the level of differential equations, as compared to algebraic splittingmethods. In the following subsections 3.1, 3.2, and 3.3 we briefly summarize the different solution strategiesconsidered in this work with a focus on the ALE relevant aspects and especially boundary conditions, whilewe refer to previous works [41, 29] for a more detailed description in the context of Eulerian formulationsand high-order DG methods. In Section 3.4, the different Navier–Stokes solvers are discussed in terms ofstability and achievable rates of convergence.

Notation Backward differentiation formula (BDF) time integration is used in this work. The time in-terval [0, T ] is divided into N time steps of variable size. With n = 0, ..., N − 1 denoting the time stepnumber, the equations are advanced from time tn to tn+1 = tn + ∆tn in time step n, leading to the time

grid tiNi=0 = t0 +∑i−1j=0 ∆tj

N

i=0. BDF schemes of order J = 1, 2, 3 are considered. Although A-stability

is only achieved for time integration schemes of order J = 1, 2, third-order accurate schemes have been foundto be useful for practical problems as well and will be investigated in this work. Coefficients γ0 and αi ofthe BDF time integration scheme as well as coefficients βi of the extrapolation scheme used to extrapolateexplicit terms are listed in Table 1 for the case of a constant time step size ∆t = T/N , see also [40]. An exten-sion to variable time step sizes is straight-forward, where the coefficients γn0 , αni , and βni vary from one timestep to the next and can be expressed as simple rational functions of the time step sizes ∆tn, ...,∆tn−J+1,see also [60]. The time integration constant for adaptive time-stepping are summarized in Appendix A.

Table 1: Coefficients of BDF time integration scheme and extrapolation scheme for constant time step size, see [40].

Order γ0 α0 α1 α2 β0 β1 β2

1 1 1 - - 1 - -2 3/2 2 −1/2 - 2 −1 -3 11/6 3 −3/2 1/3 3 −3 1

3.1. Coupled solution approach

Applying the BDF scheme to equations (5) and (6) and using a fully implicit formulation (including theconvective term), we obtain

γn0un+1 −

∑J−1i=0 α

ni u

n−i

∆tn

∣∣∣∣∣χ

+((un+1 − un+1

G

)· ∇)un+1 −∇ · Fv(un+1) +∇pn+1 = f (tn+1) , (9)

∇ · un+1 = 0 , (10)

where |χ means that all terms of the BDF sum are evaluated at constant χ, i.e., an ALE-type time derivativehas to be considered. The boundary conditions are

un+1 = gn+1u on ΓD ,(

Fv(un+1)− pn+1I)· n = hn+1 on ΓN .

6

As an alternative formulation, we also study an explicit formulation of convective term, discretized in timevia an extrapolation scheme of order J

γn0un+1 −

∑J−1i=0 α

ni u

n−i

∆tn

∣∣∣∣∣χ

+

J−1∑i=0

βni((un−i − un+1

G

)· ∇)un−i −∇ · Fv(un+1) +∇pn+1 = f (tn+1) ,

(11)

where the following boundary condition will be used in the convective term

un−i = gn+1u on ΓD . (12)

Remark While the boundary condition un−i = gn−iu might be considered a natural formulation as well, itis interesting to study whether equation (12) also preserves optimal rates of convergence. The advantageof this formulation is that only one version of the boundary condition is required at any one time of thesolution of the transient problem, therefore easing implementation without the need to store and keep trackof previous versions of the boundary condition for fluid–structure interaction problems. For the projectionmethods discussed below, we will see that we can not fully maintain this goal as these methods are moreinvolved regarding the formulation of boundary conditions.

3.2. High-order dual splitting scheme

The high-order dual splitting scheme [40] consists of the following four sub-steps to be solved in eachtime step

γn0 u−∑J−1i=0 α

ni u

n−i

∆tn

∣∣∣∣∣χ

= −J−1∑i=0

βni((un−i − un+1

G

)· ∇)un−i + f (tn+1) ,

un−i = gn+1u on ΓD,

(13)

−∇2pn+1 = − γn0∆tn∇ · u,

∇pn+1 · n = hn+1p on ΓD,

pn+1 = gn+1p on ΓN,

u = gn+1u on ΓD,

(14)

ˆu = u− ∆tnγn0∇pn+1,

pn+1 = gn+1p on ΓN,

(15)

γn0∆tn

un+1 −∇ · Fv

(un+1

)=

γn0∆tn

ˆu,

un+1 = gn+1u on ΓD,

Fv(un+1) · n = hn+1u on ΓN.

(16)

The convective term, the pressure term, and the viscous term are taken into account in different sub-steps,and the pressure Poisson equation (14) is obtained from equation (15) by requiring ∇ · ˆu = 0. The viscousterm is formulated implicitly in time, while the convective term is formulated explicitly for this scheme. Theauthors are not aware of variants of this splitting scheme that allow an implicit treatment of the convectiveterm and – at the same time – achieve higher-order accuracy in time.

The pressure Neumann boundary condition hp and the Dirichlet boundary condition gu for the intermedi-ate velocity need to be discussed in more detail. A consistent Neumann boundary condition for the pressure

7

is derived by multiplying the momentum equation of the incompressible Navier–Stokes equations by thenormal vector n [39, 40, 53], which yields the following result in case of the ALE form of the incompressibleNavier–Stokes equations

hp (tn+1) =−

γn0 gn+1u −

∑J−1i=0 α

ni g

n−iu

∆tn

∣∣∣∣∣χ

− f (tn+1)

· nn+1

−

Jp−1∑i=0

βni(((un−i − un+1

G

)· ∇)un−i + ν∇× ωn−i

) · nn+1 .

(17)

As already noted in our previous work [41] dealing with the Eulerian case, the time derivative term and bodyforce term appear on the right-hand side as well in the general case of time dependent boundary conditionsand right-hand side vectors f 6= 0, compared to the original formulation in [40] and a later work [6] wherethese terms are dropped. Compared to [41] where the exact derivative ∂gu/∂t is used by exploiting thatthis term is known analytically in the Eulerian case on ΓD, the acceleration term has to be replaced by adiscrete BDF time derivative in the ALE or fluid–structure interaction case where the boundary conditionis only known at discrete times. Hence, one has to record the history of Dirichlet boundary values gu incase of the dual splitting scheme. Since the time derivative is of ALE-type at constant χ, the convectiveterm needs to be formulated in ALE form as well. The velocity un+1

h at time tn+1 is unknown at this pointof the algorithm, so that the convective term and the viscous term need to be formulated explicitly. Forthe viscous term we use the well-known rotational formulation ∇× ω (with vorticity ω = ∇× u) which isobtained from the identity ∇2u = ∇(∇ · u)−∇× (∇× u) = −∇× (∇× u) = −∇× ω. This formulationproposed by [39, 40] makes use of the incompressibility constraint ∇ · u = 0, and it is well understood thatthe rotational formulation significantly improves accuracy as compared to the Laplace formulation [53, 52];nevertheless the latter formulation ∇2u is also used sometimes [61, 62, 38].

The velocity Dirichlet boundary condition for the intermediate velocity u is derived in a similar fashionfrom equation (13), see also [41] where this boundary condition has been proposed for the Eulerian case

gu (χ, tn+1) =

J−1∑i=0

αniγn0gu(χ, tn−i)−

∆tnγn0

J−1∑i=0

βni((un−i − un+1

G

)· ∇)un−i +

∆tnγn0

f (tn+1) , (18)

As indicated in the above equation, the history of the boundary condition gu is evaluated in grid coordi-nates χ following the moving mesh from one time instant to the next, so that the ALE form of the convectiveterm is required in this boundary condition similar to the pressure Neumann boundary condition (17). Theconvective term again needs to be extrapolated.

Remark One might raise the question why the terms in equation (17) coming from the acceleration termevaluate the prescribed boundary data gu instead of simply evaluating the numerical solution u comingfrom the interior of the domain, since this is also done for the convective and viscous terms in the boundaryconditions. From the formulations in [22, 24, 63, 62, 38] it is unclear whether the acceleration term in equa-tion (17) should directly evaluate the numerical solution u if the time derivative is not known analytically.The same holds for the BDF time derivative terms in equation (18). We found that using u instead of gu forthe time derivative terms in equations (17) and (18) leads to instabilities that occur for small time step sizes.We therefore recommend to use the available boundary data gu in the above boundary conditions wheneverpossible. We further note that the works [22, 24] do not use a boundary condition like equation (18) sincethese works use a different DG discretization of the velocity divergence term in equation (14) comparedto the present work, namely a formulation without integration by parts and, hence, without imposition ofboundary conditions, see Section 4 for details. However, it was shown in [41] that integration by parts ofthis term in the context of discontinuous Galerkin methods is essential to obtain stability for small timesteps and that a consistent Dirichlet boundary condition according to equation (18) is necessary to obtainhigh-order accuracy in time.

8

3.3. Pressure-correction scheme

Pressure-correction schemes are another class of projection methods, treating both convective and vis-cous terms in the same sub-step, see [52] for detailed information. Extending the formulation of pressure-correction schemes for non-moving meshes shown in [41, 29] to the ALE formulation of the incompressibleNavier–Stokes equations, the pressure-correction schemes can be summarized as follows

γn0 u−∑J−1i=0 α

ni u

n−i

∆tn

∣∣∣∣∣χ

+((u− un+1

G

)· ∇)u−∇ · Fv (u) =

−Jp−1∑i=0

βni ∇pn−i + f (tn+1) ,

u = gn+1u on ΓD,

Fv(u) · n = hn+1u , on ΓN,

pn−i = gn−ip on ΓN,

(19)

−∇2φn+1 = − γn0∆tn∇ · u,

∇φn+1 · n = hn+1φ = 0 on ΓD,

u = gn+1u on ΓD,

φn+1 = gn+1φ on ΓN,

(20)

pn+1 = φn+1 +

Jp−1∑i=0

βni pn−i − ζν∇ · u,

u = gn+1u on ΓD,

(21)

un+1 = u− ∆tnγn0∇φn+1,

φn+1 = gn+1φ on ΓN.

(22)

Here, Jp is the order of extrapolation of the pressure gradient term. Schemes with Jp = 0 are called non-incremental, and schemes with Jp ≥ 1 incremental pressure-correction schemes. The standard formulation isobtained for ζ = 0, while the rotational formulation corresponds to ζ = 1. We exclusively study the rotationalversion in the present work due to improved convergence rates and accuracy [52]. In the above equations, theconvective term is formulated implicitly. As for the monolithic solver, we consider an alternative formulationwith explicit treatment of the convective term, resulting in the following momentum equation in the firstsub-step

γn0 u−∑J−1i=0 α

ni u

n−i

∆tn

∣∣∣∣∣χ

−∇ · Fv (u) = −J−1∑i=0

βni((un−i − un+1

G

)· ∇)un−i −

Jp−1∑i=0

βni ∇pn−i + f (tn+1) ,

(23)

where the following boundary condition is imposed for the convective term

un−i = gn+1u on ΓD . (24)

The pressure Poisson equation (20) is subject to the pressure Dirichlet boundary condition

gφ(χ, tn+1) = gp (χ, tn+1)−Jp−1∑i=0

βigp (χ, tn−i) . (25)

9

Remark As mentioned in [41] for the Eulerian version of the present solver, the boundary condition (25) isan extension of the boundary condition (10.3) in [52] towards time-dependent pressure boundary conditionson ΓN. Note that we prescribe pn−i = gn−ip in equation (19) to be consistent with boundary condition (25).

We otherwise observe suboptimal rates of convergence when prescribing pn−i = gn+1p . Hence, a history of

pressure Dirichlet boundary values on ΓN has to be stored for the pressure-correction scheme for higher-order schemes with Jp ≥ 1, which can be seen as a consequence of the operator splitting as compared to themonolithic solver described in Section 3.1.

3.4. Discussion of incompressible Navier–Stokes solvers

Let us briefly summarize and discuss the different Navier–Stokes solver strategies. Already when consid-ering the unsteady Stokes equations without convection, projection methods are in general only conditionallystable for higher extrapolation order Jp. According to [64], the dual splitting scheme is unconditionally sta-ble for Jp ≤ 2 independent of the order 1 ≤ J ≤ 4 of the BDF scheme, but only conditionally stablefor Jp > 2. Hence, the scheme with parameters J = 3 and Jp = 2 provides the highest order of accuracy,namely ∆t3, among the schemes that are unconditionally stable. Pressure-correction schemes are only un-conditionally stable for Jp ≤ 1, while they are conditionally stable for Jp ≥ 2, see [52]. This is also inagreement with our numerical results where we observe instabilities for Jp = 2. Hence, the highest accuracycombined with unconditional stability is achieved for J = 2, Jp = 1, resulting in a second-order accuratescheme, ∆t2. Let us mention that the scheme J = 3, Jp = 2 shows indeed third-order accuracy, but isonly conditionally stable and not suited for practical problems. From these considerations, we derive theparameters Jp = min(2, J), J ≤ 3 for the dual splitting scheme, and Jp = min(2, J) − 1, J ≤ 2 for thepressure-correction scheme as an optimal choice. The monolithic solution approach and the dual splittingscheme have the advantage that the third-order schemes are stable. The fact that the pressure-correctionscheme allows both implicit and explicit formulations of the convective term can be seen as an advantageover the dual splitting scheme, especially when one wants to avoid restrictions of the time step size accord-ing to the CFL condition and when high-order of accuracy is not the primary target. Finally, the simplerstructure of algebraic equations is often considered an advantage of projection methods over the monolithicapproach. We therefore analyze these different methods in the present work since we believe they coverdifferent facets of incompressible Navier–Stokes solvers.

4. Spatial discretization

Notation The computational domain Ωh =⋃Nel

e=1 Ωe ∈ Rd is composed of Nel finite elements, which arenon-overlapping and of quadrilateral/hexahedral shape in the context of this work. The boundary Γh =∂Ωh approximates Γ, and it holds Γh = ΓD

h ∪ ΓNh with ΓD

h ∩ ΓNh = ∅ as in the spatially continuous case.

Approximations to velocity u(x, t) and pressure p(x, t) are denoted by uh(x, t) ∈ Vuh and ph(x, t) ∈ Vph,where the discontinuous Galerkin finite element spaces of test and trial functions are defined as

Vuh =uh ∈ [L2(Ωh)]

d: uh (xe(ξ, t)) |Ωe = ueh(ξ)|Ωe

∈ Vuh,e = [Qku(Ωe)]d , ∀e = 1, . . . , Nel

,

Vph =ph ∈ L2(Ωh) : ph (xe(ξ, t)) |Ωe = peh(ξ)|Ωe

∈ Vph,e = Qkp(Ωe) , ∀e = 1, . . . , Nel

.

The polynomial space Qk(Ωe) of tensor degree ≤ k is defined on the reference element Ωe = [0, 1]d withreference coordinates ξ = (ξ1, ..., ξd)

T. We approximate velocity and pressure on element e by nodal Lagrangepolynomials

ueh(ξ, t) =

ku∑i1,...,id=0

lkui1...id(ξ)uei1...id(t) , peh(ξ, t) =

kp∑i1,...,id=0

lkpi1...id

(ξ)pei1...id(t) , (26)

where uei1...id and pei1...id denote the nodal degrees of freedom of the velocity and pressure solution on

element e, respectively. The multidimensional shape functions lki1...id are given as the tensor product of

10

one-dimensional shape functions, lki1...id(ξ) =∏dn=1 l

k,1Din

(ξn), where lk,1Di (ξ) are the Lagrange polynomials

of degree k based on the Legendre–Gauss–Lobatto nodes.For reasons of inf–sup stability, the polynomial degree for the pressure is kp = ku − 1, see [41]. For ease

of notation, we simply write ku = k in the following. In the above equations, xe(ξ, t) : Ωe → Ωe(t) denotesthe mapping from reference space to physical space

fem :

Ωe × [0, T ]→ Ωe(t), Ωe = [0, 1]

d,Ωe(t) ⊂ Rd ,

(ξ, t) 7→ xe (ξ, t) .

For the mapping, the same ansatz is used as for approximating the solution, but with polynomial degree km

xe(ξ, t) =

km∑i1,...,id=0

lkmi1...id(ξ)xei1...id(t) .

The mapping can be seen in analogy to the function fG describing the topological changes of the domain Ω.While fG is defined globally for the whole domain and in a spatially continuous way, the finite elementmapping describes the mesh motion for each element of the mesh in the discrete setting and is of finitedimension. For the following derivations, it is important to realize that a point with constant χ canbe thought of as a point with fixed ξ coordinates within one element, i.e., there exists a bijective mapbetween χ and ξ for each element. An illustration is given in Figure 1. In the context of this work, werestrict ourselves to problems for which the topology of the mesh does not change, i.e., no remeshing. Hence,the data structures do not have to be adjusted dynamically when moving the mesh. Updating the meshmeans updating the d(km + 1)d mapping degrees of freedom per element, i.e.,

xei1...id(tn+1) = fG(xei1...id(t = 0), tn+1) .

Moderately large deformations are possible as long as fG remains invertible, and invalid elements withinvalid mapping or Jacobian will otherwise occur in the discrete setting. The numerical examples shown inthis work use a high-order, isoparametric mapping with km = ku.

In DG methods, integrals have to be computed over the interface fe−/e+ = ∂Ωe− ∩ ∂Ωe+ of two adja-cent elements Ωe− and Ωe+ , where the outward pointing normal vectors are n− for Ωe− and n+ for Ωe+ .Furthermore, let u−h and u+

h denote the solution uh on fe−,e+ evaluated from the interior of element e−

and element e+, respectively. By Γinth we denote the set of all interior faces. In the following, we make

use of the average operator u = (u− + u+) /2, the jump operator JuK = u− ⊗ n− + u+ ⊗ n+, andthe oriented jump operator [u] = u− − u+. Moreover, an element-by-element formulation is used wherevolume integrals are performed over the current element Ωe and face integrals over the boundary ∂Ωe ofelement e. By definition, we denote interior information on the current element Ωe by the superscript (·)−and exterior information from neighboring elements by the superscript (·)+. In this context, the normalvector n equals n−, while n+ = −n− = −n. Finally, we introduce the abbreviations (v, u)Ωe

=∫

Ωevu dΩ

and (v, u)∂Ωe=∫∂Ωe

v u dΓ, where the operator symbolizes inner products and will become clear from

the context. An integral over the computational domain is to be understood as (v, u)Ωh=∑Nel

e=1 (v, u)Ωe, and

similarly for integrals over all interior faces, e.g., (v, u∗)Γinth

=∑Nel

e=1

(12v, u

∗)∂Ωe\Γh

if u∗ is single-valued.

Due to the ansatz (26) with a separation of space and time, applying the temporal discretization to thespatially discretized equations in ALE form becomes trivial in the sense that the structure of the equationsis equivalent to the Eulerian case. Consider the time derivative term in equation (5) multiplied by test

11

functions vh, integrated over element Ωe(t = tn+1), and to be discretized in time(vh,

∂uh(t)

∂t

∣∣∣∣χ

)Ωn+1

e

=

vh, ∂∑kui1,...,id=0 l

kui1...id

(ξ)uei1...id(t)

∂t

∣∣∣∣∣ξ

Ωn+1

e

=

vh, ku∑i1,...,id=0

lkui1...id(ξ)∂uei1...id(t)

∂t

Ωn+1

e

≈

vh, ku∑i1,...,id=0

lkui1...id(ξ)γn0u

e,n+1i1...id

−∑J−1i=0 α

ni u

e,n−ii1...id

∆tn

Ωn+1

e

=

vh, γn0un+1h −

∑J−1i=0 α

ni u

n−ih

∆tn

∣∣∣∣∣ξ

Ωn+1

e

.

The BDF rule introduced in the third row of the above equation approximates the acceleration at time tn+1

consistently with the integral over Ωe taken at the same instant of time. In the following, we skip thelabel |ξ for simplicity, as it is clear from the above derivation that the BDF rule is simply applied to theglobal solution vector containing the unknown degrees of freedom and that all terms of the BDF sum use thesame mass matrix at time tn+1. As explained in more detail in Section 4.4, using the same mass matrix forall solution vectors is important in order to satisfy the geometric conservation law [15]. The above equationhighlights that discretization in space and time commutate, meaning that the last term of the above equationwould have also been obtained by discretizing equation (9) in space. However, the projection-type solutionmethods considered in this work are already formulated in a time-discrete manner, since the splitting isperformed on the level of differential operators. For this reason, the derivation of DG formulations shownin the following starts from the time-discrete problems stated in Section 3.

Following [15], the grid velocity is computed in the same way via a BDF time derivative of the nodal gridcoordinates xei1...id in the time discrete case in order to achieve high-order temporal convergence on movingmeshes

un+1G,h =

∂xh∂t

(tn+1)

∣∣∣∣χ

≈γn0 x

n+1h −

∑J−1i=0 α

ni x

n−ih

∆tn

∣∣∣∣∣ξ

. (27)

This procedure is different from [5, 6] where the grid coordinates are updated by integrating the meshvelocity forward in time. In the following, we summarize the variational formulation of the different ALENavier–Stokes solvers.

Table 2: Weak imposition of boundary conditions: choice of exterior values (·)+ on domain boundaries as a function of interiorvalues (·)− and prescribed boundary data for velocity and pressure in order to weakly impose boundary conditions accordingto the mirror principle. The procedure is equivalent to the Eulerian case [41].

ΓDh ΓN

h

velocity u+h = −u−h + 2gu u+

h = u−h∇u+

h · n = ∇u−h · n ∇u+h · n = −∇u−h · n+ 2hu

ν

pressure p+h = p−h p+

h = −p−h + 2gp∇p+

h · n = −∇p−h · n+ 2hp ∇p+h · n = ∇p−h · n

12

4.1. Monolithic solution approach

Beginning with the monolithic solution approach, the weak discontinuous Galerkin formulation of thefully discrete problem with implicit formulation of the convective term can be summarized as follows:Find un+1

h ∈ Vuh , pn+1h ∈ Vph such that(vh,

γn0un+1h −

∑J−1i=0 α

ni u

n−ih

∆tn

)Ωn+1

e

+ ce,n+1h

(vh,u

n+1h ,un+1

G,h ; gn+1u

)+ve,n+1

h

(vh,u

n+1h ; gn+1

u ,hn+1u

)+ ge,n+1

h

(vh, p

n+1h ; gn+1

p

)+ae,n+1

D,h (vh,un+1h ) + ae,n+1

C,h (vh,un+1h ; gn+1

u )− (vh,f(tn+1))Ωn+1e

= 0 , (28)

−de,n+1h (qh,u

n+1h ; gn+1

u ) = 0 , (29)

for all (vh, qh) ∈ Vuh,e×Vph,e and for all elements e = 1, ..., Nel. The time label n+ 1, e.g. in ce,n+1

h , indicates

that the integral is evaluated on the domain Ωn+1e . When formulating the convective term explicitly, the

convective term in the discretized momentum equation (28) is replaced by

ce,n+1h

(vh,u

n+1h ,un+1

G,h ; gn+1u

)→

J−1∑i=0

βni ce,n+1h

(vh,u

n−ih ,un+1

G,h ; gn+1u

).

As proposed in [49], a computational efficient variant of the monolithic system of equations, equations (28)and (29), is to apply the divergence and continuity penalty terms in a postprocessing step(

vh,un+1h

)Ωn+1

e+ ae,n+1

D,h

(vh,u

n+1h

)∆tn + ae,n+1

C,h

(vh,u

n+1h ; gn+1

u

)∆tn = (vh, uh)Ωn+1

e, (30)

where uh is an intermediate velocity obtained as the solution of the coupled system of equations withoutpenalty terms. For the numerical results studied in this work, the formulation shown in equations (28)and (29) is used with penalty terms added to the momentum equation.

We next present the discontinuous Galerkin formulation of the individual terms of the incompressibleNavier–Stokes equations, see also [41, 29] for more detailed derivations. Boundary conditions are imposedaccording to the mirror principle as summarized in Table 2. Central flux functions are used for the velocitydivergence term

deh,weak (qh,uh; gu) = − (∇qh,uh)Ωe+ (qh, uh · n)∂Ωe

= − (∇qh,uh)Ωe+ (qh, uh · n)∂Ωe\Γh

+ (qh,uh · n)∂Ωe∩ΓNh

+ (qh, gu · n)∂Ωe∩ΓDh,

(31)

and for the pressure gradient term

geh,weak (vh, ph; gp) = − (∇ · vh, ph)Ωe+ (vh, phn)∂Ωe

= − (∇ · vh, ph)Ωe+ (vh, phn)∂Ωe\Γh

+ (vh, phn)∂Ωe∩ΓDh

+ (vh, gpn)∂Ωe∩ΓNh.

(32)

As an alternative to the above weak forms, we consider the so-called strong formulations by performingintegration-by-parts once again

deh,strong (qh,uh) = (qh,∇ · uh)Ωe−(qh,

1

2[uh] · n

)∂Ωe

, (33)

geh,strong (vh, ph) = (vh,∇ph)Ωe−(vh,

1

2[ph]n

)∂Ωe

. (34)

The weak and strong formulations are equivalent as long as integrals are evaluated exactly, which doesnot hold in general for the quadrature rules typically used, see Section 4.6. For this reason, the weak

13

and strong formulations behave differently regarding the fulfillment of the geometric conservation law asdiscussed in Section 4.4. The discretization of the viscous term is based on the symmetric interior penaltyGalerkin (SIPG) method [65]

veh(vh,uh; gu,hu) = (∇vh, ν∇uh)Ωe−(∇vh,

ν

2JuhK

)∂Ωe\Γh

− (vh, ν∇uh · n)∂Ωe\Γh

+ (vh, ντJuhK · n)∂Ωe\Γh.

(35)

Inserting the boundary conditions acccording to Table 2, the viscous operator veh = veh,hom + veh,inhom can besplit into homogeneous contributions

veh,hom(vh,uh) = (∇vh, ν∇uh)Ωe−(∇vh,

ν

2JuhK

)∂Ωe\Γh

− (∇vh, νuh ⊗ n)∂Ωe∩ΓDh

− (vh, ν∇uh · n)∂Ωe\Γh− (vh, ν∇uh · n)∂Ωe∩ΓD

h

+ (vh, ντJuhK · n)∂Ωe\Γh+ (vh, 2ντuh)∂Ωe∩ΓD

h,

and inhomogeneous contributions

veh,inhom(vh; gu,hu) = (∇vh, ν gu ⊗ n)∂Ωe∩ΓDh− (vh,hu)∂Ωe∩ΓN

h− (vh, 2ντgu)∂Ωe∩ΓD

h.

The SIPG penalty parameter τ depends on the polynomial degree k and a characteristic element length h.It has to be large enough to ensure coercivity of the bilinear form. For quadrilateral/hexahedral elementsused in the present work, bounds for the penalty parameter have been derived in [66]. Following this work,the penalty parameter τe of element e is calculated as

τe = (k + 1)2A (∂Ωe \ Γh) /2 +A (∂Ωe ∩ Γh)

V (Ωe), (36)

where V (Ωe) =∫

ΩedΩ and A(f) =

∫f⊂∂Ωe

dΓ are the element volume and surface area, respectively.

The maximum value from both sides is chosen on interior faces, τ = max (τe− , τe+) if face f ⊆ ∂Ωe \ Γh,while τ = τe is used on boundary faces f ⊆ ∂Ωe ∩ Γh.

The convective term is particularly relevant in the ALE context and is written in non-conservative formwith grid velocity uG,h. We perform integration by parts twice (strong formulation), since we observedsub-optimal rates of convergence for the weak formulation in case of even polynomial degrees. Then, anupwind flux is used as numerical flux function to obtain

ceh (vh,uh,uG,h; gu) = (vh, (∇uh) · (uh − uG,h))Ωe− (vh, ((uh − uG,h) · n)uh)∂Ωe

+(vh, ((uh − uG,h) · n) uh+

1

2| (uh − uG,h) · n| [uh]︸ ︷︷ ︸

upwind flux

)∂Ωe

. (37)

To keep the formulation compact, we do not explicitly highlight the dependency of the variational form cehon the boundary condition gu as it is clear that the boundary condition enters the formulation throughthe choice of exterior values on domain boundaries according to Table 2, i.e., u+

h = −u−h + 2gu on ΓDh .

Since the convective term is nonlinear and the residual forms the right-hand side of the linear solver incase of a Newton–Krylov approach, there is no need to split the convective operator into homogeneous andinhomogeneous contributions.

Finally, the divergence penalty term aeD,h and continuity penalty term aeC,h have to be defined. Theseterms can be interpreted as a weak enforcement ofH(div)-conformity (normal continuous velocity) along withsuitable function spaces for velocity and pressure that lead to an exactly (pointwise) divergence-free velocity(such as Raviart–Thomas) [29, 36, 35]. These penalty terms are mandatory to obtain a robust discretizationfor under-resolved problems such as turbulent flows when using standard L2-conforming spaces. The penalty

14

terms are defined as [29]

aeD,h(vh,uh) = (∇ · vh, τD∇ · uh)Ωe, τD,e = ζD ‖un+1,ex

h ‖ heku + 1

,

aeC,h(vh,uh; gu) = (vh · n, τC [uh] · n)∂Ωe, τC,e = ζC ‖un+1,ex

h ‖ ,

where un+1,exh =

∑J−1i=0 β

ni u

n−ih is an extrapolation of the velocity field of order J , (·) an elementwise volume-

averaged quantity, and he = V1/3e a characteristic element length with Ve the volume of the element. A

minor modification compared to [29] is that the continuity penalty term is applied not only on interior faces,but also on boundary faces with exterior values according to Table 2. While mainly numerical examples withperiodic boundary conditions have been studied [29] where this does not make a difference, we observed thatit is advantageous to apply this penalty term on all faces for general boundary conditions. The continuitypenalty parameter is τC = τC,e on interior faces and τC = τC,e on boundary faces. As shown abovefor other operators, the continuity penalty term can then be split into homogeneous and inhomogeneouscontributions, aeC,h(vh,uh; gu) = aeC,h,hom(vh,uh) + aeC,h,inhom(vh; gu).

Remark Consistency of the above variational formulation, equations (28) and (29), immediately followsfrom the fact that the weak form is derived using integration by parts, using consistent numerical fluxfunctions, and consistent boundary conditions according to Table 2. The additional penalty terms areconsistent as well, since these terms contain the divergence of the velocity or the jump of the velocity overinterior faces.

4.2. High-order dual splitting scheme

For the dual splitting projection scheme, the variational formulation can be summarized as follows:

Find uh, ˆuh,ˆuh,u

n+1h ∈ Vuh and pn+1

h ∈ Vph such that for all vh ∈ Vuh,e, qh ∈ Vph,e and for all elements e =

1, ..., Nel(vh,

γn0 uh −∑J−1i=0 α

ni u

n−ih

∆tn

)Ωn+1

e

= −J−1∑i=0

βni ce,n+1h

(vh,u

n−ih ,un+1

G,h ; gn+1u

)+ (vh,f(tn+1))Ωn+1

e, (38)

le,n+1h,hom

(qh, p

n+1h

)= − γn0

∆tnde,n+1h

(qh, uh; gn+1

u

)− le,n+1

h,inhom

(qh; gn+1

p , hn+1p

), (39)(

vh, ˆuh

)Ωn+1

e

= (vh, uh)Ωn+1e− ∆tn

γn0ge,n+1h

(vh, p

n+1h ; gn+1

p

), (40)(

vh,γn0

∆tn

ˆuh

)Ωn+1

e

+ ve,n+1h,hom

(vh,

ˆuh

)=

(vh,

γn0∆tn

ˆuh

)Ωn+1

e

− ve,n+1h,inhom(vh; gn+1

u ,hn+1u ), (41)(

vh,un+1h

)Ωn+1

e+ ae,n+1

D,h

(vh,u

n+1h

)∆tn + ae,n+1

C,h,hom

(vh,u

n+1h

)∆tn =(

vh,ˆuh

)Ωn+1

e

− ae,n+1C,h,inhom

(vh; gn+1

u

)∆tn.

(42)

The Laplace operator leh appearing in the pressure Poisson equation is discretized using the SIPG method

leh(qh, ph; gn+1

p , hn+1p

)= (∇qh,∇ph)Ωe

−(∇qh,

1

2JphK

)∂Ωe

− (qh, ∇ph · n)∂Ωe

+ (qh, τJphK · n)∂Ωe,

(43)

and is again split into homogeneous contributions

leh,hom (qh, ph) = (∇qh,∇ph)Ωe−(∇qh,

1

2JphK

)∂Ωe\Γh

− (∇qh, phn)∂Ωe∩ΓNh

− (qh, ∇ph · n)∂Ωe\Γh− (qh,∇ph · n)∂Ωe∩ΓN

h

+ (qh, τJphK · n)∂Ωe\Γh+ (qh, 2τph)∂Ωe∩ΓN

h.

(44)

15

and inhomogeneous contributions

leh,inhom

(qh; gn+1

p , hn+1p

)=(∇qh, gpn)∂Ωe∩ΓN

h− (qh, hp)∂Ωe∩ΓD

h− (qh, 2τgp)∂Ωe∩ΓN

h. (45)

In our previous work [49], the penalty terms have been applied in the projection equation (40). How-ever, since we evaluate the continuity penalty operator also on boundary faces in the present work, thepenalty terms are evaluated in a postprocessing step, equation (42). Adding the penalty terms to the pro-jection equation (40) would prevent to achieve high-order temporal accuracy since prescribing the boundary

condition gu for the intermediate velocity ˆuh would be inconsistent.Some comments are in order regarding the evaluation of the boundary conditions gu(tn+1) and hp(tn+1)

on the right-hand side of equation (39). The convective and viscous terms have to be evaluated on ∂Ωe usingthe finite element expansion of the velocity solution uh on element e. In the discrete case, the convective termis simply calculated as (∇uh) · (uh−uG,h) by taking the derivative of the shape functions. Since the viscousterm involves second derivatives, we calculate it in a two-step process, computing the vorticity ωh ∈ Vuh ina first step by a local L2-projection, see [28, 41]

(vh,ωh)Ωe= (vh,∇× uh)Ωe

.

In the second step, the contribution of the viscous term to the pressure Neumann boundary condition isobtained by calculating the curl of the vorticity ωh according to equation (17).

4.3. Pressure-correction scheme

Finally, the variational formulation is stated for the class of pressure-correction methods: Find uh, ˆuh,un+1h ∈

Vuh and φn+1h , pn+1

h ∈ Vph such that for all vh ∈ Vuh,e, qh ∈ Vph,e and for all elements e = 1, ..., Nel(

vh,γn0 uh −

∑J−1i=0 α

ni u

n−ih

∆tn

)Ωn+1

e

+ ce,n+1h

(vh, uh,u

n+1G,h ; gn+1

u

)+ ve,n+1

h

(vh, uh; gn+1

u ,hn+1u

)= −

Jp−1∑i=0

βni geh

(vh, p

n−ih ; gn−ip

)+ (vh,f(tn+1))Ωn+1

e,

(46)

le,n+1h,hom

(qh, φ

n+1h

)= − γn0

∆tnde,n+1h

(qh, uh; gn+1

u

)− le,n+1

h,inhom

(qh; gn+1

φ , hn+1φ

), (47)

(qh, p

n+1h

)Ωn+1

e=

qh, φn+1h +

Jp−1∑i=0

(βip

n−ih

)Ωn+1

e

− ζν de,n+1h

(qh, uh; gn+1

u

), (48)

(vh, ˆuh

)Ωn+1

e

= (vh, uh)Ωn+1e− ∆tn

γn0ge,n+1h

(vh, φ

n+1h ; gn+1

φ

), (49)(

vh,un+1h

)Ωn+1

e+ ae,n+1

D,h (vh,un+1h )∆tn + ae,n+1

C,h,hom(vh,un+1h )∆tn =(

vh, ˆuh

)Ωn+1

e

− ae,n+1C,h,inhom(vh, g

n+1u )∆tn.

(50)

Similar to the monolithic solver, we consider an alternative formulation that formulates the convective termexplicitly, replacing

ce,n+1h

(vh, uh,u

n+1G,h ; gn+1

u

)→

J−1∑i=0

βni ce,n+1h

(vh,u

n−ih ,un+1

G,h ; gn+1u

).

4.4. Geometric conservation law

It can be proven that the fully discrete ALE-DG methods derived above satisfy the geometric conservationlaw, i.e., they are able to preserve a constant flow field [13]. In other words, the constant solution u(x, t) =u0, p(x, t) = p0 is a solution of the fully discrete formulations for f = 0.

16

Theorem 4.1. Assume the solution at time instant tn is given as unh = u0, pnh = p0 where ui,0 = Ci, i =1, ..., d, and p0 = C (and similarly for previous time instants in case of high-order schemes), and fur-ther assume f = 0 and exact numerical integration of the velocity divergence term and pressure gradientterm. Then, the fully discrete ALE-DG incompressible Navier–Stokes solvers introduced in Sections 4.1, 4.2,and 4.3 preserve a constant solution and yield un+1

h = u0, pn+1h = p0 at time tn+1 = tn+∆tn, independently

of the order of the time integration and extrapolation schemes, and for arbitrary mesh velocities.

Proof For the monolithic solver, equations (28) and (29), it is obvious that the time derivative term becomes

zero since it holds γn0 =∑J−1i=0 α

ni . As explained in [15], this originates from the fact that we discretize the

differential form of the ALE equations with time derivative applied to the velocity only, as opposed toformulations that apply the time derivative to an integral quantity. Hence, to complete the proof one needsto show that the weak forms of all individual terms evaluate to zero. This is trivial for the divergence andcontinuity penalty terms, since these evaluate the divergence inside the element or the jump over interiorfaces, both vanishing for a constant solution. The volume term of the convective operator contains thegradient, and the face terms vanish due to the conservativity and consistency of the numerical flux. Notethat it is enough to consider interior faces in this context, as one realizes that boundary faces behave asinterior faces when evaluating the boundary values according to Table 2 for a constant solution. The SIPGdiscretization of the viscous term also vanishes, as each form either contains gradients or jumps of thesolution. The strong formulation of the velocity divergence term, equation (33), and pressure gradient term,equation (34), vanish for constant solutions, since the volume integrals contain derivatives of the solution andthe face integrals contain jumps. The strong formulation therefore always satisfies the (discrete) geometricconservation law. Since strong and weak formulation are only equivalent under the assumption of exactintegration, these terms do not vanish exactly for the weak formulation of the velocity–pressure couplingterms, equation (31) and (32), when applying standard Gaussian quadrature rules and considering arbitrarilydeformed elements. Since the divergence and continuity penalty terms vanish for a constant solution, theGCL is also fulfilled when applying these terms in a postprocessing step, equation (30).

Regarding the dual splitting scheme, it follows from the above argumentation that the first sub-stepin equation (38) yields uh = u0. Particular attention has to be paid to the boundary conditions hpin equation (17) and gu in equation (18). The pressure Neumann boundary condition vanishes for f =0, as it contains derivatives in either space or time, and we obtain gu = gu in the case of a constantsolution. Hence, the divergence operator on the right-hand side of the pressure Poisson equation (39)vanishes, and the pressure Poisson equation is satisfied for a constant solution p0. It immediately followsfrom the argumentation for the monolithic solver that the remaining sub-steps, equations (40), (41), and (42),

yield ˆuh = u0,ˆuh = u0, and un+1

h = u0, thus preserving a constant flow state.Turning to the pressure-correction scheme, we first notice that the momentum equation (46) results

in uh = u0 for reasons explained above. The pressure boundary condition gφ becomes zero, as∑Jp−1i=0 βni = 1,

so that φn+1h = 0 is a solution of the pressure Poisson equation (47). The pressure update equation (48)

results in pn+1h = p0, since the divergence term on the right-hand side becomes zero. The same holds for the

pressure gradient term in equation (49), since its arguments are φn+1h = 0, gn+1

φ = 0, so that the constant

solution ˆuh = u0 and un+1h = u0 is recovered in the last sub-steps.

Remark It is worth noting that no assumption has been made regarding the mesh velocity or how itis computed numerically to show compliance with the GCL. This is also a consequence of the fact thatwe discretize – as suggested in [15] – the differential form of the ALE equations in convective formulation,equation (5). In Section 5, we demonstrate numerically that the geometric conservation is fulfilled exactly forthe strong formulations, and that it is not fulfilled exactly down to rounding errors for the weak formulationof the velocity–pressure coupling terms in general. However, since the temporal discretization and spatialdiscretization are designed to satisfy the GCL, we expect that no relevant difference is observed for practicalproblems (where the solution is non-constant) due to this variational crime, since those problems alwayssuffer from non-exact integration, and variational crimes have to be accepted in several respects. Forexample, fulfilling discrete energy stability exactly with respect to the velocity–pressure coupling terms, seeSection 4.5, requires that one term is formulated in weak form, and the other one in strong form, so that the

17

formulation becomes symmetric independently of integration errors. We also emphasize that previous workshave shown that fulfilling the GCL is neither a necessary nor a sufficient condition for the time integratorto preserve its high-order accuracy on moving meshes [16, 67]. We will therefore carefully investigate thetemporal convergence behavior of the present ALE schemes and demonstrate that the high-order accuracyof the Navier–Stokes solvers on fixed meshes is preserved on moving meshes when using definition (27) tocalculate the mesh velocity.

4.5. Energy stability

As detailed in the introduction, energy stability is a crucial ingredient to obtain a flow solver that isrobust in the under-resolved regime. In this section, we analyze the energy balance of the present ALE-DGmethods. For this analysis, we assume vanishing body forces, f = 0, and consider the inviscid limit, ν = 0.This is reasonable since the viscous term has a dissipative character both physically and numerically, wherethe numerically dissipative character immediately follows from the fact that the SIPG discretization of theviscous term, equation (35), is positive definite and symmetric. Therefore, the critical case is to investigatewhether a numerical method is energy stable in the absence of viscous dissipation. For simplicitly, we alsoassume periodic boundaries (including periodicity of the grid velocity). Under these assumptions and forsmooth solutions, the ALE incompressible Navier–Stokes equations (5) and (6) are energy-conserving in thefollowing sense

∂

∂t

∫Ω0

1

2u · udetJ dΩ

∣∣∣∣χ

=

∫Ω(t)

∂ 12u · u∂t

∣∣∣∣χ

dΩ +

∫Ω(t)

1

2(u · u)∇ · uG dΩ = 0 , (51)

where J = ∂x/∂χ is the Jacobian. A derivation of this relation is shown in Appendix B. A spatiallydiscretized ALE incompressible Navier–Stokes solver(

vh,∂uh∂t

∣∣∣∣χ

)Ωe

+ ceh (vh,uh,uG,h) + geh (vh, ph) + aeD,h(vh,uh) + aeC,h(vh,uh) = 0 , (52)

−deh(qh,uh) = 0 , (53)

is called energy-stable if it fulfills the following discrete analogy∫Ωh(t)

∂ 12uh · uh∂t

∣∣∣∣χ

dΩ +

Nel∑e=1

∫Ωe(t)

1

2(uh · uh)∇ · uG,hdΩ ≤ 0 . (54)

We begin with reformulating the left term in equation (54) so that the semi-discrete momentum equation (52)can be inserted∫

Ωh(t)

∂ 12uh · uh∂t

∣∣∣∣χ

dΩ =

∫Ωh(t)

uh ·∂uh∂t

∣∣∣∣χ

dΩ =

Nel∑e=1

(uh,

∂uh∂t

∣∣∣∣χ

)Ωe

= −Nel∑e=1

(ceh (uh,uh,uG,h) + geh (uh, ph) + aeD,h(uh,uh) + aeC,h(uh,uh)

). (55)

Regarding the pressure gradient term, we first observe that the following relation holds

Nel∑e=1

geh,strong (uh, ph) = (uh,∇ph)Ωh− (uh, JphK)Γint

h= −

Nel∑e=1

deh,weak (ph,uh) = 0 , (56)

and similarly between geh,weak and deh,strong. For these combinations of the velocity–pressure coupling terms,the formulation is symmetric independently of integration errors, and the pressure gradient term will notcontribute to the energy evolution due to the discrete continuity equation (53).

18

The divergence and continuity penalty terms have a dissipative character as these are positive semi-definite by definition. Hence, it remains to consider the convective term. Inserting equation (37) andreformulating yields

Nel∑e=1

ceh (uh,uh,uG,h) = +

Nel∑e=1

((uh, (∇uh) · (uh − uG,h))Ωe

− (uh, ((uh − uG,h) · n)uh)∂Ωe

)+

Nel∑e=1

(uh, ((uh − uG,h) · n) uh+

1

2| (uh − uG,h) · n| [uh]

)∂Ωe

= +

Nel∑e=1

((uh, (∇uh) · (uh − uG,h))Ωe

−(uh, ((uh − uG,h) · n)

1

2[uh]

)∂Ωe

)

+

([uh],

1

2| (uh − uG,h) · n| [uh]

)Γinth

.

(57)

In Appendix C, we show that the first row on the right-hand side of the above equation can be reformulatedas follows by algebraic manipulations

Nel∑e=1

((uh, (∇uh) · (uh − uG,h))Ωe

−(uh, ((uh − uG,h) · n)

1

2[uh]

)∂Ωe

)=

= −1

2(∇ · (uh − uG,h),uh · uh)Ωh

+1

2([uh] · n, uh · uh)Γint

h,

(58)

which is easier to interpret in terms of energy stability since this formulation contains the divergence ofthe velocity, i.e., a residual of the incompressible Navier–Stokes equations, and the divergence of the gridvelocity. It is worth emphasizing that the face integrals related to the moving mesh dropped out completely,since the ALE term is a linear transport term. The volume integral of the ALE transport term does notdrop out since the grid velocity is not divergence-free. However, one can see that this is exactly the secondterm in (54). Inserting equations (56), (57), and (58) into equation (55) yields the following result∫

Ωh(t)

∂ 12uh · uh∂t

∣∣∣∣χ

dΩ +

Nel∑e=1

∫Ωe(t)

1

2(uh · uh)∇ · uG,hdΩ =

= +1

2(∇ · uh,uh · uh)Ωh

− aD,h(uh,uh)

− 1

2([uh] · n, uh · uh)Γint

h− aC,h(uh,uh)

−(

[uh],1

2| (uh − uG,h) · n| [uh]

)Γinth

.

(59)

It is interesting to realize that the result derived in [29] for the conservative formulation of the convectiveterm in the Eulerian case is very similar to the convective formulation in ALE form considered here, whereterms with exactly the same structure occur. In particular, one observes that the ALE formulation does notintroduce new terms on the right-hand side as compared to the Eulerian case (a consequence of the fact thatthe additional ALE term is a linear transport term). One might therefore argue that energy stability forthe Eulerian case translates into energy stability for the ALE case with moving meshes. The third row onthe right-hand side is the upwind stabilization term of the convective term and always exhibits a dissipativebehavior. However, it is well-known that this term is not able to render the nonlinear convective termenergy-stable. As argued in [29, 36], the consistent divergence and continuity penalty terms are positivesemi-definite and control the non-vanishing divergence and non-vanishing jumps of the velocity in normaldirection. By the use of consistent penalty terms, energy stability of the DG discretization is enforced

19

weakly; there is currently no proof that the kinetic energy is strictly non-increasing at all times. Thesign-indefinite terms of the above equation are exactly zero for an H(div)-conforming (normal continuous)velocity space together with a pressure space that ensures that the velocity is pointwise divergence-free. Forthese reasons, the stabilized approach with divergence and continuity penalty terms can also be denotedas H(div)-stabilization.

4.6. Numerical integration and implementation

Integrals in the variational form are evaluated numerically by means of Gaussian quadrature, wherewe choose the number of one-dimensional quadrature points to ensure exact integration on affine elementgeometries with constant Jacobian. The velocity mass matrix term, the body force term, the viscousterm, the velocity divergence term, the pressure gradient term, and the two penalty terms are integratedwith nq = ku+1 quadrature points. To avoid aliasing effects, we use nq = b 3ku

2 c+1 quadrature points for theconvective term containing quadratic nonlinearities. The Laplace operator in the pressure Poisson equationand the pressure mass matrix operator are integrated with nq = kp + 1 quadrature points. The presentALE-DG methods are implemented in C++ using the deal.II finite element library [68], and especiallythe matrix-free evaluation techniques developed in [51] for the evaluation of volume and face integrals ofdiscretized DG operators. State-of-the-art iterative solvers are used to solve the (non-)linear systems ofequations of the fully discrete problem.

4.7. CFL condition

For the formulations with explicit treatment of the convective term, the time step size is restrictedaccording to the CFL condition. Since the transport velocity is uh − uG,h in the ALE case, the CFLcondition used on static meshes has to be adjusted accordingly using the relative velocity between fluid andgrid motion. Here, we distinguish between two types of CFL condition. The global CFL condition [21, 29]applied to the ALE case

∆t =Cr

k1.5u

hmin

‖uh − uG,h‖max, (60)

with global estimates of the minimum element length hmin and maximum velocity ‖uh−uG,h‖max is used fortime stepping with constant ∆t. In the above equation, Cr denotes the Courant number and the term k1.5

u

was found to describe well the relation between critical time step size and polynomial degree for the presentDG discretization [29]. Since the minimum element length and the maximum velocity are difficult to estimatea priori and since the maximum velocity does not necessarily occur in the smallest element, a local CFLcondition is used in case of adaptive time-stepping [69]

∆t = mine=1,...,Nel

∆te, ∆te = minq=1,...,Nq,e

Cr

k1.5u

h

‖uh − uG,h‖

∣∣∣∣q,e

, (61)

with the local velocity-to-mesh-size ratio‖uh−uG,h‖

h

∣∣∣q,e

= ‖J−T(uh − uG,h)‖q,e evaluated at quadrature

point q of element e. This CFL condition ensures that the time step size does not exceed the critical one inany element in any quadrature point.

5. Numerical results

The aim of this section is to display the numerical discretization properties of the proposed ALE-DGincompressible Navier–Stokes solvers. We select a set of academic test cases that address different aspectsof ALE solvers, with the goal to obtain a picture as complete as possible. In detail, we study the geometricconservation property by the example of the free stream preservation test in Section 5.1. The convergencebehavior in terms of temporal as spatial convergence rates are investigated in Section 5.2 by the exampleof a two-dimensional vortex problem with moving Dirichlet and Neumann boundaries. In this section,

20

we also test the robustness of the different incompressible Navier–Stokes solvers in the limit of large griddeformations. Finally, the robustness of the proposed discretization methods for under-resolved turbulentflows is studied in Section 5.3 by considering the three-dimensional Taylor–Green vortex problem for viscousflows at Re = 1600, and also in the very challenging inviscid limit.

Convergence rates and relative L2-errors for problems with known analytical solution are computed asdefined in [41]. Solver tolerances are selected to not spoil accuracy, e.g., by choosing relative solver tolerancesof 10−6 and absolute solver tolerances of 10−12. In case of pure Dirichlet boundary conditions, the pressurelevel is undefined and can be fixed, e.g., by setting the mean value of the pressure DoF vector to zero.

5.1. Geometric conservation law – free stream preservation test

We study the free stream preservation test to investigate whether the ALE formulations derived abovefulfill the geometric conservation law. Satisfying the geometric conservation law means that a constant flowfield is not disturbed by a moving mesh, i.e., the solver is able to preserve the free stream flow conditionsexactly and independently of the mesh deformation. The analytical solution of the free stream preservationtest is therefore the constant flow state

u(x, t) = (1, . . . , 1)T, p(x, t) = 1 ,

where we prescribe pure Dirichlet boundary conditions u(x, t) = gu(x, t) on ΓDh (t) = Γh(t). The computa-

tional domain at initial time is Ω0 = Ω(t = 0) = [−L/2, L/2]2

with L = 1. The simulation is run over atime interval of 0 ≤ t ≤ T = 10. The following analytical mesh movement with sine functions in both timeand space is prescribed in two space dimensions

x(χ, t) = χ+A sin

(2π

t

TG

)sin(

2π χ2+L/2L

)sin(

2π χ1+L/2L

) , (62)

and in three space dimensions

x(χ, t) = χ+A sin

(2π

t

TG

)sin(

2π χ2+L/2L

)sin(

2π χ3+L/2L

)sin(

2π χ1+L/2L

)sin(

2π χ3+L/2L

)sin(

2π χ1+L/2L

)sin(

2π χ2+L/2L

) , (63)

where the amplitude is set to A = 0.08 resulting in a strongly deformed mesh. The period length of thegrid motion is set to TG = T/10 and the wavenumber in space is chosen such that the length and heightof the domain are exactly one period. In Figure 2(b), the mesh deformation is illustrated for d = 2, wherethe initial, undeformed mesh is a uniform Cartesian grid. The mesh reaches its maximum deformation attimes t = TG/4 + i TG/2, i = 0, 1, 2, .... The viscosity is set to ν = 0.025. Adaptive time-stepping is usedwhere the time step size is adjusted dynamically according to the CFL condition (61) using Cr = 0.25. Inthe following, we use a mesh with 8d elements as in Figure 2 and consider a polynomial degree of k = 3.Both absolute and relative solver tolerances are set to a small value of 10−14. Table 3 reports relativeerrors for velocity and pressure for BDF schemes of order 1 to 3 and the three different incompressibleNavier–Stokes solvers considered in this work. For the dual splitting scheme, Jp = min(2, J) is used butthe simulations have also been stable for the choice Jp = J for the high-order BDF scheme J = 3. For thepressure-correction scheme, Jp = min(2, J)− 1 is used for all J and the rotational formulation with χ = 1.Here, the choice Jp = J − 1 lead to instabilities for the high-order scheme J = 3 in agreement with theory.

The results in Table 3 reveal that all schemes fulfill the geometric conservation law for d = 2, and inparticular also for the weak formulation of the velocity–pressure coupling terms. For d = 3, the geometricconservation law is fulfilled exactly only when using the strong formulations deh,strong and geh,strong as expectedtheoretically, see Section 4.4. Errors larger than the solver tolerances are observed for the weak formulationsfor d = 3. For all variants studied here, similar results are obtained when using an implicit formulation of

21

(a) undeformed mesh (b) deformed mesh (A/L = 0.08) (c) theoretical limit Amax/L = 12π

Figure 2: Illustration of sine-like mesh motion in two space dimensions according to equation (62).

the convective term for the monolithic solver and the pressure-correction scheme, where we had to slightlyrelax the solver tolerances to 10−12 to ensure convergence of the Newton solver. Hence, it remains to explainwhy the weak formulation of velocity–pressure coupling terms fulfills the GCL exactly for d = 2. As thecritical aspect in this context is the exact evaluation of integrals, see Section 4.4, it can be conjectured thatintegrals of the velocity–pressure coupling terms are indeed evaluated exactly for d = 2 for constant solutionson deformed elements. However, this is a special case that only occurs for the free stream preservation testdue to a solution of lowest polynomial degree, and this does not hold for general non-constant solutions andarbitrarily deformed elements. Hence, we do not pay further attention to this point. In a similar direction,our results show that the weak formulations are very accurate as well, and will therefore be used for thefollowing examples. The reason for this choice is that the conclusion is even stronger when being able todemonstrate optimal convergence behavior for a formulation that appears to be sub-optimal regarding thefree stream preservation test.

5.2. Temporal and spatial convergence behavior

Next, we analyze the convergence behavior of the ALE-DG methods and study whether optimal rates ofconvergence observed in the Eulerian case carry over to moving meshes. For this purpose, we select the two-dimensional vortex problem from [22], which is an analytical solution of the two-dimensional incompressibleNavier–Stokes equations in the absence of body forces, f = 0, namely

u(x, t) =

(− sin(2πx2)+ sin(2πx1)

)exp

(−4νπ2t

),

p(x, t) = − cos(2πx1) cos(2πx2) exp(−8νπ2t

),

(64)

with viscosity set to ν = 0.025, and simulated over the time interval 0 ≤ t ≤ T = 1. The computationaldomain at start time is Ω0 = [−L/2, L/2]2 with length L = 1, and is deformed according to the two-dimensional mesh movement function (62) with parameters A = 0.08 and TG = 4T (maximum deformationreached at end time t = T ) unless specified otherwise. Again, a mesh as depicted in Figure 2(b) withrefinement level l is used. Sine-like mesh deformations are commonly used to verify high-order ALE-DGimplementations, see for example [9, 10, 12]. In these works, however, the sine functions are defined in away that the boundaries are not moving. Instead, we intentionally choose a setup for which the boundariesare moving since our goal is to test all parts of the algorithm relevant for FSI. The verification of boundaryconditions is particularly relevant for the splitting-type solvers and some effects might not be visible if theboundaries are fixed. For example, if the boundaries are non-moving, the normal vector in equation (17)would not change over time and the ALE transport term would simply drop out since uG = 0 on the

22

Table 3: Numerical results for free stream preservation test for both strong and weak formulations of velocity–pressure couplingterms and two- and three-dimensional problems: Jp = min(2, J) is used for the dual splitting scheme, and Jp = min(2, J)− 1for the pressure-correction scheme in incremental formulation. An explicit formulation of the convective term is used for allthree solvers.

(a) Two-dimensional problem (d = 2), weak formulations deh,weak and geh,weak

relative L2-error uh relative L2-error phBDF1 BDF2 BDF3 BDF1 BDF2 BDF3

monolithic 1.8E–16 1.8E–15 2.1E–15 1.3E–16 6.0E–13 2.9E–13

dual splitting 1.2E–15 1.6E–15 1.8E–15 2.5E–13 7.5E–14 6.1E–13

pressure-correction 1.6E–15 2.7E–15 3.0E–15 1.8E–13 9.5E–14 4.6E–13

(b) Two-dimensional problem (d = 2), strong formulations deh,strong and geh,strong

relative L2-error uh relative L2-error phBDF1 BDF2 BDF3 BDF1 BDF2 BDF3

monolithic 1.8E–16 1.1E–15 1.8E–15 1.3E–16 2.1E–13 2.4E–13

dual splitting 1.0E–15 1.3E–15 1.4E–15 1.5E–13 3.8E–13 6.5E–13

pressure-correction 1.0E–15 1.6E–15 1.8E–15 2.3E–13 2.3E–13 4.7E–13

(c) Three-dimensional problem (d = 3), weak formulations deh,weak and geh,weak

relative L2-error uh relative L2-error phBDF1 BDF2 BDF3 BDF1 BDF2 BDF3

monolithic 1.3E–08 1.3E–08 1.3E–08 8.7E–09 8.2E–09 8.2E–09

dual splitting 1.1E–08 1.2E–08 1.2E–08 6.4E–09 6.6E–09 6.8E–09

pressure-correction 1.2E–08 1.3E–08 1.3E–08 7.0E–09 7.1E–08 6.1E–08

(d) Three-dimensional problem (d = 3), strong formulations deh,strong and geh,strong

relative L2-error uh relative L2-error phBDF1 BDF2 BDF3 BDF1 BDF2 BDF3

monolithic 3.5E–16 1.4E–14 1.2E–14 1.7E–16 8.5E–13 8.4E–13

dual splitting 3.5E–16 1.8E–15 1.1E–15 1.4E–13 7.3E–13 1.3E–12

pressure-correction 1.9E–15 1.9E–15 2.3E–15 3.3E–13 7.8E–13 1.1E–12

boundary. According to the setup in [22], each of the four sides of the square is split into a Dirichletboundary and a Neumann boundary according to the inflow and outflow sections, respectively. Note thatthe chosen mesh deformation is in compliance with these boundary conditions.

Figure 3 shows a visualization of the solution at the time of maximal mesh deformation t = T usingpolynomial shape functions of degree k = 3 and considering the two lowest refinement levels of l = 1, 2 (thegrid has to consist of at least 22 elements due to the type of boundary conditions prescribed with each faceof the rectangular domain cut into a Dirichlet part and a Neumann part). While the velocity field is alreadywell resolved on the coarsest mesh with l = 1, the pressure field of polynomial degree 2 is approximatedpoorly for refinement level l = 1 with distinct discontinuities between the elements. For l = 2, the pressuresolution appears to be visually converged with only minor differences as compared to the solution on evenfiner meshes. In the following, we study the convergence quantitatively in terms of errors against theanalytical solution as well as convergence rates measured in space and time.

In a first set of experiments, we test the temporal convergence behavior for both constant and adaptive

23

(a) p(x, T ) (l = 1, k = 3) (b) p(x, T ) (l = 2, k = 3) (c) ‖u(x, T )‖ (l = 1, k = 3) (d) ‖u(x, T )‖ (l = 2, k = 3)

Figure 3: Vortex problem: visualization of solution at final time t = T for two different mesh resolutions, l = 1 and l = 2, withpolynomial degree k = 3 for the velocity and 2 for the pressure (red indicates high values and blue low values). The amplitudeof the mesh deformation is A = 0.08.

time step sizes. The chosen spatial resolution is fine, l = 3 and k = 8, to make sure that errors are dominatedby temporal discretization errors. The second aspect why this test case is interesting is the fact that theCFL condition does often not show up for this particular test case for moderate Reynolds numbers, probablydue to the fact that the vortex is not moving. This allows to measure temporal convergence rates of anincompressible Navier–Stokes solver with explicit formulation of the convective term, which would be difficultotherwise because in the regular case with Cr < Crcrit one is often operating in a regime where temporaldiscretization errors are already negligible as compared to spatial errors for higher-order time integrationschemes.

Figure 4 shows results of a temporal convergence study for BDF schemes of order J = 1, 2, 3 usingconstant and adaptive time step sizes. All types of solvers converge with optimal rates of convergence onthe moving mesh. Compared to additional simulations performed for a static, Cartesian mesh, the errorsare almost the same and only slightly larger. The lowest errors are obtained for the monolithic solver withimplicit formulation of the convective term. The dual splitting scheme and the monolithic solver with explicitconvective term yield similar errors, and the errors are again slightly larger for both explicit and implicitpressure-correction formulations. For very small time step sizes and the BDF3 scheme, the spatial errorbecomes dominant at some point. Note that BDF3 schemes are not considered for the pressure-correctionscheme since Jp = 1 (required for stability) limits convergence rates to second-order in that case. For thedual splitting scheme, large errors occur for Courant numbers Cr > 1. This effect does not show up forthe non-moving mesh and we conjecture that this effect originates from the CFL condition. For a regularNavier–Stokes problem with explicit formulation of the convective term, no stability can be expected inthis range of Courant numbers. While a sharp CFL bound is not visible for the chosen parameters, allsolvers become unstable for Cr > 1 (for J = 2) when using a smaller viscosity of ν = 10−3 (higher Reynoldsnumber), showing the usual sharp CFL bound.

In a second set of experiments, we study the spatial convergence behavior for polynomial degrees k =2, 3, 4, 5 by a mesh refinement study, considering refine levels l = 1, ..., 6. The BDF2 time integration schemewith a small, constant time step of ∆t = 5 · 10−5 is used to obtain small temporal errors. Figure 5 showsresults for the three different solver types with an explicit formulation of the convective term. Since thisis a spatial convergence test, the results would be indistinguishable when using an implicit formulation ofthe convective term and are therefore not shown explicitly. For comparison, we also show results for a non-moving, Cartesian mesh for the monolithic solution approach. Overall, all variants converge will optimalrates of convergence for all polynomial degrees until the temporal discretization error is reached. Comparedto the static mesh, the errors are slightly larger on the moving mesh, and the gap between moving and staticmeshes increases for increasing polynomial degree.

Finally, we test the robustness by increasing the amplitude A of the mesh movement function to itstheoretical limit, i.e., the value at which the aspect ratio tends to infinity or at which inverted elementsoccur, in order to characterize the point where the proposed ALE-DG methods will break down. This

24

(a) constant ∆t

(b) adaptive ∆tn

Figure 4: Vortex problem: temporal convergence tests for ALE incompressible Navier–Stokes solvers for BDF schemes oforder J = 1, ..., 3 with Jp = min(2, J) for the dual splitting scheme and Jp = min(2, J) − 1, J ≤ 2 for the pressure-correctionscheme.

theoretical limit is reached when the lower left corner of the domain becomes an arbitrarily thin needle, seeFigure 2(c). Mathematically, this limit is reached when the slope of the lower domain boundary reaches avalue of 1 in the lower left corner at the time of maximum deformation, i.e.,

∂x2(χ1, χ2 = −L/2, t = TG/4)

∂χ1

∣∣∣∣χ1=−L/2

= A cos

(2πχ1 + L/2

L

)∣∣∣∣χ1=−L/2

2π

L

!= 1

Amax

L=

1

2π.

In Figure 6, we plot the relative errors of velocity and pressure for a coarse mesh with 42 elements andpolynomial degree k = 3 as a function of A/Amax. Adaptive time-stepping, equation (61), with Cr = 0.2 isused. The error increases moderately for small amplitudes of the mesh deformation, and a rapid increase inerrors can be observed around 85% of the theoretically maximum amplitude for this problem.

25

Figure 5: Vortex problem: spatial convergence tests for ALE incompressible Navier–Stokes solvers for polynomial degrees k =2, 3, 4, 5 and comparison to static Cartesian mesh.

Figure 6: Vortex problem: robustness test of ALE incompressible Navier–Stokes solvers on a mesh with 42 elements andpolynomial degree k = 3.

5.3. Robustness for under-resolved turbulent flows

We study the applicability of the present ALE-DG solvers to transitional and turbulent flows by theexample of the three-dimensinal Taylor–Green vortex (TGV) problem [70]. Starting from the initial state

u1(x, t = 0) = + sin (x1) cos (x2) cos (x3) ,

u2(x, t = 0) = − cos (x1) sin (x2) cos (x3) ,

u3(x, t = 0) = 0 ,

p(x, t = 0) =1

16(cos (2x1) + cos (2x2)) (cos (2x3) + 2) ,

the flow transitions to turbulence in the absence of body forces, f = 0. The Reynolds number is Re = 1/ν,and we consider both the standard setting Re = 1600 and the inviscid limit Re → ∞ in the present work.The simulated time interval is 0 ≤ t ≤ T = 20. At start time, the domain Ω0 = [−L/2, L/2]

3= [−π, π]

3is

a Cartesian box that deforms over time according to the mesh motion described in equation (63), with anamplitude of A = π/6 and varying mesh velocities characterized by period times decreassing from TG = 20

26

to TG = 1. Periodic boundaries are used in all coordinate directions. The domain boundaries are movingfor the given mesh motion, but the mesh deformation is defined periodically in order to ensure consistencywith the periodic boundary conditions. An illustration of the mesh deformation for the above parametersis given in Figure 7. We consider meshes that are originally Cartesian with Nel = (2l)3 elements, where ldenotes the level of refinement, and denote the number of velocity nodes (2l(k + 1))3 as effective meshresolution. A BDF2 time integration scheme along with an explicit treatment of the convective term is usedfor all solver types, with Jp = 2 for the dual splitting scheme and Jp = 1 for the pressure-correction schemeunless specified otherwise. Moreover, we use adaptive time-stepping, equation (61), with a Courant numberof Cr = 0.2.

(a) ‖u(x, t = 0)‖ (b) ‖u(x, t = 5)‖ (c) ‖u(x, t = 10)‖ (d) ‖u(x, t = 15)‖ (e) ‖u(x, t = 20)‖

Figure 7: Taylor–Green vortex problem at Re = 1600: visualization of velocity magnitude at different times for a spatialresolution with l = 3 and polynomial degree k = 7. The parameters of the mesh deformation are A = π/6 and TG = 20. Thecolor map has been rescaled for each time instant, where red indicates high velocity and blue low velocity. The results shownhave been simulated with the dual splitting scheme.

In a first set of experiments, the viscous case at Re = 1600 is investigated. In Figure 8, results for thekinetic energy Ek =

∫Ωh

12uh · uhdΩ/

∫Ωh

dΩ and the dissipation rate of the kinetic energy obtained on themoving mesh are compared to results on a static Cartesian mesh. A mesh refinement study for degree k = 3is performed using a rather slow mesh motion with a period time of TG = T . The results converge towardsthe accurate DNS reference solution under mesh refinement, and the solution quality is comparable for staticand moving meshes. In Figure 9, we test the robustness w.r.t. the grid velocity for the 643 mesh resolutionby decreasing the period of the mesh motion down to TG = T/20 = 1, resulting in a very fast mesh motion.With increasing mesh velocity, an oscillating behavior can be observed in the kinetic energy dissipationrate where the frequency of these oscillations follows the mesh motion. However, the temporal evolution ofthe kinetic energy is almost indistinguishable for the different mesh velocities. It can be observed that theoscillations in the dissipation rate are very small in the beginning of the simulation where the solution issmooth, while the oscillations grow once the transition to a turbulent state took place. In Section 4.5, itwas noted that the ALE transport term does not contribute to the energy evolution apart from the upwindstabilization term. However, this only holds under the assumption of exact numerical integration, whichis not fulfilled on generally deformed geometries. A possible explanation for the results in Figure 9 couldtherefore be that integration errors (which are larger for non-smooth solutions or under-resolved scenarios)are amplified if the period of the mesh motion TG tends to zero and the mesh velocity tends to infinity. Thisis supported by the observation that the oscillications are larger on coarser meshes. Let us mention thatthis experiment is performed here to test the robustness of the solver and that such a scenario (increasingthe mesh velocity for a fixed fluid velocity) is not representative of a fluid–structure interaction problem forwhich the fluid has to follow the motion of the fluid–structure interface due to no-slip conditions.

In a second set of experiments, the inviscid Taylor–Green vortex problem is studied, which is consideredone of the most challenging benchmark examples to test the robustness of a flow solver for turbulent flowsdue to the absence of viscous dissipation, see for example [12]. Although the test case is academic, it canbe expected that if a numerical method is robust for the inviscid Taylor–Green problem, it can also besuccessfully applied to practical, engineering problems. Table 4 shows results of robustness tests for theinviscid limit, where polynomial degrees of k = 3, 5, 7, 11, 15 are considered for several spatial refinement

27

(a) Kinetic energy.

(b) Kinetic energy dissipation rate.

Figure 8: Taylor–Green vortex problem at Re = 1600: comparison between static and moving meshes for polynomial degree k =3 for increasing mesh resolutions of 323, 643, and 1283. The results shown have been simulated with the dual splitting scheme.

levels l. The highest mesh velocity studied above (corresponding to TG = 1) is used for the inviscidTGV simulations. Robustness is particularly critical for coarse spatial resolutions where the flow is severlyunder-resolved, see for example [29] for similar considerations on static meshes and a comparison of thepresent stabilized DG approach to a non-stabilized one, which is why we start our investigations with amesh consisting of only one element, l = 0. With the standard penalty factor of ζ = 1 for the divergenceand continuity penalty terms, robustness is achieved for all spatial resolutions and for all solver types(monolithic, dual splitting, pressure-correction). There is one exception: For the pressure-correction schemeand the parameters described above, using the incremental formulation lead to instabilities for the coarsestpossible spatial resolution, l = 0 and k = 3, while the simulation was again stable for the non-incrementalformulation (Jp = 0). Overall, these results are encouraging in the sense that the stabilized DG approachproposed in [29] for static meshes is well designed, meaning that robustness carries over to moving mesheswithout having to adjust discretization parameters. At the same time, we emphasize again that it is unclearto which extent energy stability can be guaranteed theoretically for the present stabilized DG approach.

28

(a) Kinetic energy.

(b) Kinetic energy dissipation rate.

Figure 9: Taylor–Green vortex problem at Re = 1600: comparison between static and moving meshes for polynomial degree k =3 for increasing mesh velocity (decreasing period times of TG = 20, 4, 2, 1). A mesh with polynomial degree k = 3 and effectiveresolution of 643 is considered. The results shown have been simulated with the dual splitting scheme.

6. Conclusion and outlook

We presented ALE-DG methods for the incompressible Navier–Stokes equations that are up to third-orderaccurate in time and arbitrarily high-order accurate in space for sufficiently smooth problems. Moving meshformulations are derived for both monolithic and splitting approaches based on a method-of-lines approach,considering both implicit and explicit formulations of the convective term. The time integration frameworkrelies on BDF and extrapolation schemes and extends naturally to adaptive time-stepping. Stable and high-order accurate boundary conditions are derived for the splitting-type approaches. The ALE methods aredesigned to automatically fulfill the geometric conservation law. An important aspect is that the proposedmethods are simple to implement since the equations are solved on the deformed geometry, i.e., the genericfinite element software takes care of the mapping and the geometry terms, and only one instance of themesh is stored at a time. Fast matrix-free evaluation techniques are applied for all parts of the Navier–Stokes solvers as in the Eulerian case. A key feature is the use of consistent divergence and continuitypenalty terms to stabilize the method for under-resolved turbulent flows. Numerical results demonstrateoptimality in terms of convergence rates and the geometric conservation law, and robustness and accuracy

29

Table 4: Inviscid 3D Taylor–Green vortex problem: robustness of proposed ALE-DG methods is tested for refinement levels l =

0, ..., 5 and polynomial degrees k = 3, 5, 7, 11, 15. All simulations completed successfully as indicated by the symbol!. The sign’−’ indicates that the specific spatial resolution is not considered since the effective resolution is limited to coarse discretizationsfor this robustness test.

Polynomial degree k

l k = 3 k = 5 k = 7 k = 11 k = 15

0 ! ! ! ! !

1 ! ! ! ! !

2 ! ! ! ! !

3 ! ! ! ! −4 ! ! ! − −5 ! − − − −

of the proposed methods has been investigated for under-resolved turbulent flows. In the future, applicationto fluid–structure interaction problems is planned.

Appendix A. Adaptive time-stepping

For variable time step sizes, the time integration constants can be derived from Lagrange interpolationpolynomials. To obtain the BDF coefficients γn0 , αni , the derivative of the Lagrange interpolation poly-nomials with support points at tn+1, tn, ..., tn−J+1 is evaluated at time tn+1. To obtain the extrapolationcoefficients βni , the Lagrange interpolation polynomials with support points at tn, ..., tn−J+1 are evaluatedat time tn+1. Table A.5 summarizes the time integration constants for variable time step sizes using thenotation introduced in Section 3.

Table A.5: Coefficients of BDF time integration scheme and extrapolation scheme for adaptive time step sizes.

BDF1 BDF2 BDF3

γn0 1 2∆tn+∆tn−1

∆tn+∆tn−11 + ∆tn

∆tn+∆tn−1+ ∆tn

∆tn+∆tn−1+∆tn−2

αn0 1 ∆tn+∆tn−1

∆tn−1

(∆tn+∆tn−1)(∆tn+∆tn−1+∆tn−2)∆tn−1(∆tn−1+∆tn−2)

αn1 - − ∆t2n(∆tn+∆tn−1)∆tn−1

− ∆t2n(∆tn+∆tn−1+∆tn−2)(∆tn+∆tn−1)∆tn−1∆tn−2

αn2 - -∆t2n(∆tn+∆tn−1)

(∆tn+∆tn−1+∆tn−2)(∆tn−1+∆tn−2)∆tn−2

βn0 1 ∆tn+∆tn−1

∆tn−1

(∆tn+∆tn−1)(∆tn+∆tn−1+∆tn−2)∆tn−1(∆tn−1+∆tn−2)

βn1 - − ∆tn∆tn−1

−∆tn(∆tn+∆tn−1+∆tn−2)∆tn−1∆tn−2

βn2 - - ∆tn(∆tn+∆tn−1)(∆tn−1+∆tn−2)∆tn−2

Appendix B. Energy conservation property on moving meshes

In this section, we derive an energy conservation property according to equation (51) for the ALE form ofthe incompressible Navier–Stokes equations under the assumption of ν = 0, f = 0, and periodic boundaries.

30

We begin with ∫Ω(t)

∂ 12u · u∂t

∣∣∣∣χ

dΩ =

∫Ω(t)

u · ∂u∂t

∣∣∣∣χ

dΩ

= −∫

Ω(t)

u · (((u− uG) · ∇)u+∇p) dΩ ,

(B.1)

where the momentum equation in ALE form, equation (5), has been inserted in the second step. Next, theconvective term is reformulated by making use of the identity u ·∇u ·w = − 1

2 (u · u)∇·w+ 12∇· (w (u · u))

with w = u− uG, so that we obtain after applying Gauss’ divergence theorem for the second term∫Ω(t)

u · ∇u · (u− uG) dΩ =− 1

2

∫Ω(t)

(u · u)∇ · (u− uG) dΩ +1

2

∫∂Ω(t)

(u · u) (u− uG) · n dΓ

= +1

2

∫Ω(t)

(u · u)∇ · uG dΩ .

(B.2)

In the second step, we exploited that the surface integral vanishes due to the assumption of periodicity, andthat ∇·u = 0 holds (continuity equation). For the pressure gradient term, integration by parts is performedto obtain ∫

Ω(t)

u∇p dΩ = −∫

Ω(t)

∇ · u︸ ︷︷ ︸=0

p dΩ +

∫∂Ω(t)

p u · n dΓ = 0 , (B.3)

where the volume integral vanished due to ∇ · u = 0 and the surface integral due to periodic boundaries.Inserting equations (B.2) and (B.3) into equation (B.1), we arrive at the result∫

Ω(t)

∂ 12u · u∂t

∣∣∣∣χ

dΩ +

∫Ω(t)

1

2(u · u)∇ · uG dΩ = 0 .

To explain why this equation describes conservation of kinetic energy, we transform the integral onto thereference domain Ω0, using the Jacobian J = ∂x/∂χ∫

Ω0

∂ 12u · u∂t

∣∣∣∣χ

detJdΩ +

∫Ω0

1

2(u · u)∇ · uG detJ dΩ = 0 .

Using the relation

∂ detJ

∂t

∣∣∣∣χ

= ∇ · uG detJ ,

which is frequently used in the context of ALE derivations, see for example [15], we can further simplify theintegral

0 =

∫Ω0

∂ 12u · u∂t

∣∣∣∣χ

detJ +1

2(u · u)

∂ detJ

∂t

∣∣∣∣χ

dΩ =

∫Ω0

∂ 12u · udetJ

∂t

∣∣∣∣χ

dΩ =

=∂

∂t

∫Ω0

1

2u · u detJ dΩ

∣∣∣∣χ

.

In the last step, the time derivative has been pulled out of the integral since the integral is performedover Ω0, which does not depend on t. From the above relation, it is obvious that energy is conserved underthe given assumptions.

31

Appendix C. Energy balance of convective term

In this section, we show that equation (58) holds.

Proof The first goal is to achieve that the volume integral of the convective term contains the divergenceof the velocity and the grid velocity. Similar to the continuous case in Appendix B, we make use of theidentity

uh · ∇uh ·wh = −1

2∇ ·wh (uh · uh) +

1

2∇ · (wh (uh · uh)) ,

with wh = uh−uG,h. Note that we cannot exploit that the divergence of the discrete velocity uh is zero inthe discrete case. Inserting the above identity into the left-hand side of equation (58) and applying Gauss’divergence theorem yields

Nel∑e=1

((uh, (∇uh) · (uh − uG,h))Ωe

−(uh, ((uh − uG,h) · n)

1

2[uh]

)∂Ωe

)

=−Nel∑e=1

(1

2(∇ · (uh − uG,h),uh · uh)Ωe

)

+

Nel∑e=1

(1

2(uh · uh, (uh − uG,h) · n)∂Ωe

−(uh, ((uh − uG,h) · n)

1

2[uh]

)∂Ωe

)

=− 1

2(∇ · (uh − uG,h),uh · uh)Ωh

−1

2([uh · uh],uG,h · n)Γint

h+ ([uh] · uh,uG,h · n)Γint

h︸ ︷︷ ︸=0 (linear transport)

+1

2

(u−h · n

−,u−h · u−h

)Γinth

+1

2

(u+h · n

+,u+h · u

+h

)Γinth

− ([uh] · uh, uh · n)Γinth

,

(C.1)

where the summation over all elements has been performed in the second step. It can easily be verified thatthe surface integrals related to the grid velocity add up to zero, as expected for a linear transport term. Thethird row on the right-hand side of the above equation is more difficult to summarize, since none of the twoslots is the same for both sides of the face (originating from the nonlinearity of the transport term), so we

32

need to insert a relation like u+h = u−h − [uh] in order to be able to combine these terms

+1

2

(u−h · n

−,u−h · u−h

)Γinth

+1

2

(u+h · n

+,u+h · u

+h

)Γinth

− ([uh] · uh, uh · n)Γinth

= +1

2

(u−h · u

−h ,u

−h · n

−)Γinth

− 1

2

(u+h · u

−h ,u

−h · n

−)Γinth︸ ︷︷ ︸

combine

+1

2

(u+h · [uh],u−h · n

−)Γinth

+1

2

(u+h · u

−h , [uh] · n−

)Γinth

− 1

2

(u+h · [uh], [uh] · n−

)Γinth︸ ︷︷ ︸

combine

− ([uh] · uh, uh · n)Γinth

= +1

2

([uh] · u−h ,u

−h · n

−)Γinth

+1

2

(u+h · [uh],u−h · n

−)Γinth︸ ︷︷ ︸

combine

+1

2

(u+h · u

+h , [uh] · n−

)Γinth

− ([uh] · uh, uh · n)Γinth

= +

([uh] ·

u−h + u+h

2,u−h · n

−)

Γinth

− ([uh] · uh, uh · n)Γinth︸ ︷︷ ︸

combine

+1

2

(u+h · u

+h , [uh] · n−

)Γinth

= +([uh] · uh,

(u−h − uh

)· n−

)Γinth

+1

2

(u+h · u

+h , [uh] · n−

)Γinth︸ ︷︷ ︸

combine

= +1

2

([uh] · uh+ u+

h · u+h , [uh] · n−

)Γinth

= +1

2

(uh · uh, [uh] · n−

)Γinth

,

(C.2)

where one can easily verify that [uh] · uh + u+h · u

+h = uh · uh. Inserting equation (C.2) into

equation (C.1) completes the proof.

Acknowledgments

The research presented in this paper was partly funded by the German Research Foundation (DFG) underthe project “High-order discontinuous Galerkin for the exa-scale” (ExaDG) within the priority program“Software for Exascale Computing” (SPPEXA), grant agreement no. KR4661/2-1 and WA1521/18-1.

References

[1] C. Hirt, A. Amsden, J. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, Journal of Com-putational Physics 14 (3) (1974) 227 – 253. doi:10.1016/0021-9991(74)90051-5.

[2] J. Donea, S. Giuliani, J. Halleux, An arbitrary Lagrangian–Eulerian finite element method for transient dynamic fluid-structure interactions, Computer Methods in Applied Mechanics and Engineering 33 (1) (1982) 689 – 723. doi:10.1016/

0045-7825(82)90128-1.[3] J. Donea, A. Huerta, J.-P. Ponthot, A. Rodrguez-Ferran, Arbitrary Lagrangian–Eulerian Methods, American Cancer

Society, 2017, pp. 1–23. doi:10.1002/9781119176817.ecm2009.[4] J. Donea, P. Fasoli-Stella, S. Giuliani, Lagrangian and Eulerian finite element techniques for transient fluid-structure

interaction problems, Therm and Fluid/Struct Dyn Anal.[5] T. J. Hughes, W. K. Liu, T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous

flows, Computer Methods in Applied Mechanics and Engineering 29 (3) (1981) 329 – 349. doi:10.1016/0045-7825(81)

90049-9.[6] A. Beskok, T. C. Warburton, An unstructured hp finite-element scheme for fluid flow and heat transfer in moving domains,

Journal of Computational Physics 174 (2) (2001) 492 – 509. doi:10.1006/jcph.2001.6885.[7] M. Lesoinne, C. Farhat, Geometric conservation laws for flow problems with moving boundaries and deformable meshes,

and their impact on aeroelastic computations, Computer Methods in Applied Mechanics and Engineering 134 (1) (1996)71 – 90. doi:10.1016/0045-7825(96)01028-6.

33

[8] I. Lomtev, R. Kirby, G. Karniadakis, A discontinuous Galerkin ALE method for compressible viscous flows in movingdomains, Journal of Computational Physics 155 (1) (1999) 128 – 159. doi:10.1006/jcph.1999.6331.

[9] V.-T. Nguyen, An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for simulations of flows over variablegeometries, Journal of Fluids and Structures 26 (2) (2010) 312 – 329. doi:10.1016/j.jfluidstructs.2009.11.002.

[10] D. J. Mavriplis, C. R. Nastase, On the geometric conservation law for high-order discontinuous Galerkin discretizations ondynamically deforming meshes, Journal of Computational Physics 230 (11) (2011) 4285 – 4300, special issue High OrderMethods for CFD Problems. doi:10.1016/j.jcp.2011.01.022.

[11] P.-O. Persson, J. Bonet, J. Peraire, Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains,Computer Methods in Applied Mechanics and Engineering 198 (2009) 1585–1595. doi:10.1016/j.cma.2009.01.012.

[12] G. Schnucke, N. Krais, T. Bolemann, G. J. Gassner, Entropy stable discontinuous Galerkin schemes on moving mesheswith summation-by-parts property for hyperbolic conservation laws, arXiv preprint arXiv:1812.09093.

[13] P. D. Thomas, C. K. Lombard, Geometric conservation law and its application to flow computations on moving grids,AIAA Journal 17 (10) (1979) 1030–1037. doi:10.2514/3.61273.

[14] C. Farhat, P. Geuzaine, Design and analysis of robust ALE time-integrators for the solution of unsteady flow problemson moving grids, Computer Methods in Applied Mechanics and Engineering 193 (39) (2004) 4073 – 4095, the ArbitraryLagrangian-Eulerian Formulation. doi:10.1016/j.cma.2003.09.027.

[15] C. Forster, W. A. Wall, E. Ramm, On the geometric conservation law in transient flow calculations on deforming domains,International Journal for Numerical Methods in Fluids 50 (12) (2006) 1369–1379. doi:10.1002/fld.1093.

[16] S. Etienne, A. Garon, D. Pelletier, Perspective on the geometric conservation law and finite element methods for ALEsimulations of incompressible flow, Journal of Computational Physics 228 (7) (2009) 2313 – 2333. doi:10.1016/j.jcp.

2008.11.032.[17] B. Cockburn, G. Kanschat, D. Schotzau, A locally conservative LDG method for the incompressible Navier–Stokes equa-

tions, Mathematics of Computation 74 (251) (2005) 1067–1095.[18] B. Cockburn, G. Kanschat, D. Schotzau, A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes

equations, Journal of Scientific Computing 31 (1) (2007) 61–73. doi:10.1007/s10915-006-9107-7.[19] B. Cockburn, G. Kanschat, D. Schotzau, An equal-order DG method for the incompressible Navier–Stokes equations,

Journal of Scientific Computing 40 (1-3) (2009) 188–210.[20] F. Bassi, A. Crivellini, D. D. Pietro, S. Rebay, An artificial compressibility flux for the discontinuous Galerkin solution of

the incompressible Navier–Stokes equations, Journal of Computational Physics 218 (2) (2006) 794 – 815. doi:10.1016/j.jcp.2006.03.006.

[21] K. Shahbazi, P. F. Fischer, C. R. Ethier, A high-order discontinuous Galerkin method for the unsteady incompressibleNavier–Stokes equations, Journal of Computational Physics 222 (1) (2007) 391 – 407. doi:10.1016/j.jcp.2006.07.029.

[22] J. S. Hesthaven, T. Warburton, Nodal discontinuous Galerkin methods: Algorithms, analysis, and applications, Springer,2007. doi:10.1007/978-0-387-72067-8.

[23] L. Botti, D. A. D. Pietro, A pressure-correction scheme for convection-dominated incompressible flows with discontinuousvelocity and continuous pressure, Journal of Computational Physics 230 (3) (2011) 572 – 585. doi:10.1016/j.jcp.2010.

10.004.[24] E. Ferrer, R. H. J. Willden, A high order discontinuous Galerkin finite element solver for the incompressible Navier–Stokes

equations, Computers & Fluids 46 (1) (2011) 224 – 230. doi:10.1016/j.compfluid.2010.10.018.[25] B. Klein, F. Kummer, M. Oberlack, A SIMPLE based discontinuous Galerkin solver for steady incompressible flows,

Journal of Computational Physics 237 (2013) 235 – 250. doi:10.1016/j.jcp.2012.11.051.[26] D. T. Steinmoeller, M. Stastna, K. G. Lamb, A short note on the discontinuous Galerkin discretization of the pressure

projection operator in incompressible flow, Journal of Computational Physics 251 (2013) 480 – 486. doi:10.1016/j.jcp.

2013.05.036.[27] S. M. Joshi, P. J. Diamessis, D. T. Steinmoeller, M. Stastna, G. N. Thomsen, A post-processing technique for stabilizing

the discontinuous pressure projection operator in marginally-resolved incompressible inviscid flow, Computers & Fluids139 (2016) 120 – 129, 13th USNCCM International Symposium of High-Order Methods for Computational Fluid Dynamics- A special issue dedicated to the 60th birthday of Professor David Kopriva. doi:10.1016/j.compfluid.2016.04.021.

[28] B. Krank, N. Fehn, W. A. Wall, M. Kronbichler, A high-order semi-explicit discontinuous Galerkin solver for 3D incom-pressible flow with application to DNS and LES of turbulent channel flow, Journal of Computational Physics 348 (2017)634–659. doi:10.1016/j.jcp.2017.07.039.

[29] N. Fehn, W. A. Wall, M. Kronbichler, Robust and efficient discontinuous Galerkin methods for under-resolved turbulentincompressible flows, Journal of Computational Physics 372 (2018) 667–693. doi:10.1016/j.jcp.2018.06.037.

[30] C. Lehrenfeld, J. Schoberl, High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady in-compressible flows, Computer Methods in Applied Mechanics and Engineering 307 (2016) 339 – 361. doi:10.1016/j.cma.2016.04.025.

[31] M. Piatkowski, S. Mthing, P. Bastian, A stable and high-order accurate discontinuous Galerkin based splitting methodfor the incompressible Navie–Stokes equations, Journal of Computational Physics 356 (2018) 220 – 239. doi:10.1016/j.

jcp.2017.11.035.[32] P. W. Schroeder, G. Lube, Divergence-free H(div)-FEM for time-dependent incompressible flows with applications

to high Reynolds number vortex dynamics, Journal of Scientific Computing 75 (2) (2018) 830–858. doi:10.1007/

s10915-017-0561-1.[33] S. Rhebergen, G. N. Wells, A hybridizable discontinuous Galerkin method for the Navier–Stokes equations with pointwise

divergence-free velocity field, Journal of Scientific Computing 76 (3) (2018) 1484–1501. doi:10.1007/s10915-018-0671-4.[34] P. W. Schroeder, G. Lube, Stabilised dG-FEM for incompressible natural convection flows with boundary and moving

34

interior layers on non-adapted meshes, Journal of Computational Physics 335 (2017) 760 – 779. doi:10.1016/j.jcp.2017.01.055.

[35] M. Akbas, A. Linke, L. G. Rebholz, P. W. Schroeder, The analogue of grad–div stabilization in DG methods for incom-pressible flows: Limiting behavior and extension to tensor-product meshes, Computer Methods in Applied Mechanics andEngineering 341 (2018) 917 – 938. doi:10.1016/j.cma.2018.07.019.

[36] N. Fehn, M. Kronbichler, C. Lehrenfeld, G. Lube, P. W. Schroeder, High-order DG solvers for under-resolved turbulentincompressible flows: A comparison of L2 and H(div) methods, International Journal for Numerical Methods in Fluids inpress (2019) n/a. doi:10.1002/fld.4763.

[37] E. Ferrer, R. H. J. Willden, A high order discontinuous Galerkin Fourier incompressible 3D Navier-Stokes solver withrotating sliding meshes, Journal of Computational Physics 231 (21) (2012) 7037 – 7056. doi:10.1016/j.jcp.2012.04.039.

[38] Y. Wang, A. Quaini, S. Canic, A higher-order discontinuous Galerkin/arbitrary Lagrangian Eulerian partitioned ap-proach to solving fluid–structure interaction problems with incompressible, viscous fluids and elastic structures, Journalof Scientific Computing 76 (1) (2018) 481–520. doi:10.1007/s10915-017-0629-y.

[39] S. A. Orszag, M. Israeli, M. O. Deville, Boundary conditions for incompressible flows, J. Sci. Comput. 1 (1) (1986) 75–111.doi:10.1007/BF01061454.

[40] G. E. Karniadakis, M. Israeli, S. A. Orszag, High-order splitting methods for the incompressible Navier–Stokes equations,Journal of Computational Physics 97 (2) (1991) 414 – 443. doi:10.1016/0021-9991(91)90007-8.

[41] N. Fehn, W. A. Wall, M. Kronbichler, On the stability of projection methods for the incompressible Navier–Stokesequations based on high-order discontinuous Galerkin discretizations, Journal of Computational Physics 351 (2017) 392–421. doi:10.1016/j.jcp.2017.09.031.

[42] L. Xu, X. Xu, X. Ren, Y. Guo, Y. Feng, X. Yang, Stability evaluation of high-order splitting method for incompress-ible flow based on discontinuous velocity and continuous pressure, Advances in Mechanical Engineering 11 (10) (2019)1687814019855586. doi:10.1177/1687814019855586.

[43] S. Rhebergen, B. Cockburn, A space–time hybridizable discontinuous Galerkin method for incompressible flows on de-forming domains, Journal of Computational Physics 231 (11) (2012) 4185 – 4204. doi:10.1016/j.jcp.2012.02.011.

[44] T. L. Horvath, S. Rhebergen, A locally conservative and energy-stable finite-element method for the Navier–Stokes problemon time-dependent domains, International Journal for Numerical Methods in Fluids 89 (12) (2019) 519–532. doi:10.1002/fld.4707.

[45] C. Cantwell, D. Moxey, A. Comerford, A. Bolis, G. Rocco, G. Mengaldo, D. D. Grazia, S. Yakovlev, J.-E. Lombard,D. Ekelschot, B. Jordi, H. Xu, Y. Mohamied, C. Eskilsson, B. Nelson, P. Vos, C. Biotto, R. Kirby, S. Sherwin, Nektar++:An open-source spectral/hp element framework, Computer Physics Communications 192 (2015) 205 – 219. doi:10.1016/j.cpc.2015.02.008.

[46] P. Fischer, S. Kerkemeier, A. Peplinski, D. Shaver, A. Tomboulides, M. Min, A. Obabko, E. Merzari, NEK5000 Web page,https://nek5000.mcs.anl.gov (2020).

[47] G. J. Gassner, A. D. Beck, On the accuracy of high-order discretizations for underresolved turbulence simulations, Theo-retical and Computational Fluid Dynamics 27 (3) (2013) 221–237. doi:10.1007/s00162-011-0253-7.

[48] R. Moura, G. Mengaldo, J. Peiro, S. Sherwin, On the eddy-resolving capability of high-order discontinuous Galerkinapproaches to implicit LES / under-resolved DNS of Euler turbulence, Journal of Computational Physics 330 (2017) 615– 623. doi:10.1016/j.jcp.2016.10.056.

[49] N. Fehn, W. A. Wall, M. Kronbichler, Efficiency of high-performance discontinuous Galerkin spectral element methodsfor under-resolved turbulent incompressible flows, International Journal for Numerical Methods in Fluids 88 (1) (2018)32–54. doi:10.1002/fld.4511.

[50] M. Kronbichler, W. A. Wall, A Performance Comparison of Continuous and Discontinuous Galerkin Methods with FastMultigrid Solvers, SIAM Journal on Scientific Computing 40 (5) (2018) A3423–A3448. doi:10.1137/16M110455X.

[51] M. Kronbichler, K. Kormann, Fast matrix-free evaluation of discontinuous Galerkin finite element operators, ACM Trans.Math. Softw. 45 (3) (2019) 29:1–29:40. doi:10.1145/3325864.

[52] J. L. Guermond, P. Minev, J. Shen, An overview of projection methods for incompressible flows, Computer Methods inApplied Mechanics and Engineering 195 (4447) (2006) 6011 – 6045. doi:10.1016/j.cma.2005.10.010.

[53] G. E. Karniadakis, S. J. Sherwin, Spectral/hp element methods for computational fluid dynamics, Oxford UniversityPress, 2013. doi:10.1093/acprof:oso/9780198528692.001.0001.

[54] A. J. Chorin, Numerical solution of the Navier–Stokes equations, Math. Comp. 22 (104) (1968) 745–762. doi:10.1090/

S0025-5718-1968-0242392-2.[55] C. Hirt, J. Cook, Calculating three-dimensional flows around structures and over rough terrain, Journal of Computational

Physics 10 (2) (1972) 324–340.[56] K. Goda, A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows,

Journal of Computational Physics 30 (1) (1979) 76–95.[57] J. Van Kan, A second-order accurate pressure-correction scheme for viscous incompressible flow, SIAM Journal on Scientific

and Statistical Computing 7 (3) (1986) 870–891.[58] L. Timmermans, P. Minev, F. Van De Vosse, An approximate projection scheme for incompressible flow using spectral

elements, International Journal for Numerical Methods in Fluids 22 (7) (1996) 673–688.[59] J. Guermond, J. Shen, On the error estimates for the rotational pressure-correction projection methods, Mathematics of

Computation 73 (248) (2004) 1719–1737.[60] D. Wang, S. J. Ruuth, Variable step-size implicit-explicit linear multistep methods for time-dependent partial differential

equations, Journal of Computational Mathematics (2008) 838–855.[61] J. A. Escobar-Vargas, P. J. Diamessis, T. Sakai, A spectral quadrilateral multidomain penalty method model for high

35

Reynolds number incompressible stratified flows, International Journal for Numerical Methods in Fluids 75 (6) (2014)403–425. doi:10.1002/fld.3899.

[62] P. Gao, J. Ouyang, P. Dai, W. Zhou, A coupled continuous and discontinuous finite element method for the incompressibleflows, International Journal for Numerical Methods in Fluids 84 (8) (2017) 477–493. doi:10.1002/fld.4358.

[63] E. Ferrer, D. Moxey, R. H. J. Willden, S. J. Sherwin, Stability of projection methods for incompressible flows using highorder pressure-velocity pairs of same degree: Continuous and discontinuous Galerkin formulations, Commun. Comput.Phys. 16 (2014) 817–840. doi:10.4208/cicp.290114.170414a.

[64] E. Leriche, E. Perchat, G. Labrosse, M. O. Deville, Numerical evaluation of the accuracy and stability properties ofhigh-order direct Stokes solvers with or without temporal splitting, J. Sci. Comput. 26 (1) (2006) 25–43. doi:10.1007/

s10915-004-4798-0.[65] D. N. Arnold, F. Brezzi, B. Cockburn, L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic

problems, SIAM Journal on Numerical Analysis 39 (5) (2002) 1749–1779.[66] K. Hillewaert, Development of the discontinuous Galerkin method for high-resolution, large scale CFD and acoustics in

industrial geometries, Ph.D. thesis, Univ. de Louvain (2013).[67] P. Geuzaine, C. Grandmont, C. Farhat, Design and analysis of ALE schemes with provable second-order time-accuracy

for inviscid and viscous flow simulations, Journal of Computational Physics 191 (1) (2003) 206 – 227. doi:10.1016/

S0021-9991(03)00311-5.[68] G. Alzetta, D. Arndt, W. Bangerth, V. Boddu, B. Brands, D. Davydov, R. Gassmoeller, T. Heister, L. Heltai, K. Kormann,

M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin, D. Wells, The deal.II library, version 9.0, J. Numer. Math. 26 (4)(2018) 173–184. doi:10.1515/jnma-2018-0054.URL www.dealii.org

[69] G. E. Karniadakis, S. J. Sherwin, Spectral/hp element methods for computational fluid dynamics, 2nd Edition, OxfordUniversity Press, 2005.

[70] G. Taylor, A. Green, Mechanism of the production of small eddies from large ones, Proceedings of the Royal Society ofLondon. Series A, Mathematical and Physical Sciences 158 (895) (1937) 499–521.

36